Chapter 5

Try It

5.1 Quadratic Functions

The path passes through the origin and has vertex at $(- 4, 7),$ so $h (x) = - \frac{7}{16} {(x + 4)}^{2} + 7.$ To make the shot, $h (- 7.5)$ would need to be about 4 but $h (- 7.5) \approx 1.64;$ he doesn’t make it.

$g (x) = x^{2} - 6 x + 13$ in general form; $g (x) = {(x - 3)}^{2} + 4$ in standard form

The domain is all real numbers. The range is $f (x) \geq \frac{8}{11},$ or $[\frac{8}{11}, \infty) .$

y-intercept at (0, 13), No $x -$ intercepts

ⓐ3 seconds
ⓑ256 feet
ⓒ7 seconds

5.2 Power Functions and Polynomial Functions

$f (x)$ is a power function because it can be written as $f (x) = 8 x^{4} .$ The other functions are not power functions.

As $x$ approaches positive or negative infinity, $f (x)$ decreases without bound: as $x \to \pm \infty, f (x) \to - \infty$ because of the negative coefficient.

The degree is 6. The leading term is $- x^{6} .$ The leading coefficient is $- 1.$

As $x \to \infty, f (x) \to - \infty; a s x \to - \infty, f (x) \to - \infty .$ It has the shape of an even degree power function with a negative coefficient.

The leading term is $0.2 x^{3},$ so it is a degree 3 polynomial. As $x$ approaches positive infinity, $f (x)$ increases without bound; as $x$ approaches negative infinity, $f (x)$ decreases without bound.

y-intercept $(0, 0);$ x-intercepts $(0, 0), (- 2, 0),$ and $(5, 0)$

There are at most 12 $x -$ intercepts and at most 11 turning points.

The end behavior indicates an odd-degree polynomial function; there are 3 $x -$ intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.

The $x -$ intercepts are $(2, 0), (- 1, 0),$ and $(5, 0),$ the y-intercept is $(0, 2),$ and the graph has at most 2 turning points.

5.3 Graphs of Polynomial Functions

y-intercept $(0, 0);$ x-intercepts $(0, 0), (- 5, 0), (2, 0),$ and $(3, 0)$

The graph has a zero of –5 with multiplicity 3, a zero of -1 with multiplicity 2, and a zero of 3 with multiplicity 4.

Because $f$ is a polynomial function and since $f (1)$ is negative and $f (2)$ is positive, there is at least one real zero between $x = 1$ and $x = 2.$

$f (x) = - \frac{1}{8} {(x - 2)}^{3} {(x + 1)}^{2} (x - 4)$

The minimum occurs at approximately the point $(0, - 6.5),$ and the maximum occurs at approximately the point $(3.5, 7) .$

5.4 Dividing Polynomials

$4 x^{2} - 8 x + 15 - \frac{78}{4 x + 5}$

$3 x^{3} - 3 x^{2} + 21 x - 150 + \frac{1, 090}{x + 7}$

$3 x^{2} - 4 x + 1$

5.5 Zeros of Polynomial Functions

$f (- 3) = - 412$

The zeros are 2, –2, and –4.

There are no rational zeros.

The zeros are $–4, \frac{1}{2}, and 1 .$

$f (x) = - \frac{1}{2} x^{3} + \frac{5}{2} x^{2} - 2 x + 10$

There must be 4, 2, or 0 positive real roots and 0 negative real roots. The graph shows that there are 2 positive real zeros and 0 negative real zeros.

3 meters by 4 meters by 7 meters

5.6 Rational Functions

End behavior: as $x \to \pm \infty, f (x) \to 0;$ Local behavior: as $x \to 0, f (x) \to \infty$ (there are no x- or y-intercepts)

Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.

The function and the asymptotes are shifted 3 units right and 4 units down. As $x \to 3, f (x) \to \infty,$ and as $x \to \pm \infty, f (x) \to - 4.$

The function is $f (x) = \frac{1}{{(x - 3)}^{2}} - 4.$

$\frac{12}{11}$

The domain is all real numbers except $x = 1$ and $x = 5.$

Removable discontinuity at $x = 5.$ Vertical asymptotes: $x = 0, x = 1.$

Vertical asymptotes at $x = 2$ and $x = - 3;$ horizontal asymptote at $y = 4.$

For the transformed reciprocal squared function, we find the rational form. $f (x) = \frac{1}{{(x - 3)}^{2}} - 4 = \frac{1 - 4 {(x - 3)}^{2}}{{(x - 3)}^{2}} = \frac{1 - 4 (x^{2} - 6 x + 9)}{(x - 3) (x - 3)} = \frac{- 4 x^{2} + 24 x - 35}{x^{2} - 6 x + 9}$

Because the numerator is the same degree as the denominator we know that as $x \to \pm \infty, f (x) \to - 4; so y = - 4$ is the horizontal asymptote. Next, we set the denominator equal to zero, and find that the vertical asymptote is $x = 3,$ because as $x \to 3, f (x) \to \infty .$ We then set the numerator equal to 0 and find the x-intercepts are at $(2.5, 0)$ and $(3.5, 0) .$ Finally, we evaluate the function at 0 and find the y-intercept to be at $(0, \frac{- 35}{9}) .$

Horizontal asymptote at $y = \frac{1}{2} .$ Vertical asymptotes at $x = 1 and x = 3.$ y-intercept at $(0, \frac{4}{3} .)$

x-intercepts at $(2, 0) and (- 2, 0) .$ $(- 2, 0)$ is a zero with multiplicity 2, and the graph bounces off the x-axis at this point. $(2, 0)$ is a single zero and the graph crosses the axis at this point.

Graph of f(x)=(x+2)^2(x-2)/2(x-1)^2(x-3) with its vertical and horizontal asymptotes.

5.7 Inverses and Radical Functions

$f^{- 1} (f (x)) = f^{- 1} (\frac{x + 5}{3}) = 3 (\frac{x + 5}{3}) - 5 = (x - 5) + 5 = x$ and $f (f^{- 1} (x)) = f (3 x - 5) = \frac{(3 x - 5) + 5}{3} = \frac{3 x}{3} = x$

$f^{- 1} (x) = x^{3} - 4$

$f^{- 1} (x) = \sqrt{x - 1}$

$f^{- 1} (x) = \frac{x^{2} - 3}{2}, x \geq 0$

$f^{- 1} (x) = \frac{2 x + 3}{x - 1}$

5.8 Modeling Using Variation

$\frac{128}{3}$

$\frac{9}{2}$

$x = 20$

5.1 Section Exercises

When written in that form, the vertex can be easily identified.

If $a = 0$ then the function becomes a linear function.

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

$g (x) = {(x + 1)}^{2} - 4,$ Vertex $(- 1, - 4)$

$f (x) = {(x + \frac{5}{2})}^{2} - \frac{33}{4},$ Vertex $(- \frac{5}{2}, - \frac{33}{4})$

11.

$f (x) = 3 {(x - 1)}^{2} - 12,$ Vertex $(1, - 12)$

13.

$f (x) = 3 {(x - \frac{5}{6})}^{2} - \frac{37}{12},$ Vertex $(\frac{5}{6}, - \frac{37}{12})$

15.

Minimum is $- \frac{17}{2}$ and occurs at $\frac{5}{2} .$ Axis of symmetry is $x = \frac{5}{2} .$

17.

Minimum is $- \frac{17}{16}$ and occurs at $- \frac{1}{8} .$ Axis of symmetry is $x = - \frac{1}{8} .$

19.

Minimum is $- \frac{7}{2}$ and occurs at $−3.$ Axis of symmetry is $x = −3.$

21.

Domain is $(- \infty, \infty) .$ Range is $[2, \infty) .$

23.

Domain is $(- \infty, \infty) .$ Range is $[−5, \infty) .$

25.

Domain is $(- \infty, \infty) .$ Range is $[−12, \infty) .$

27.

$f (x) = x^{2} + 4 x + 3$

29.

$f (x) = x^{2} - 4 x + 7$

31.

$f (x) = - \frac{1}{49} x^{2} + \frac{6}{49} x + \frac{89}{49}$

33.

$f (x) = x^{2} - 2 x + 1$

35.

Vertex: (3, −10), axis of symmetry: x = 3, intercepts: $(3 + \sqrt{10}, 0)$ and $(3 - \sqrt{10}, 0)$

A parabola on a Cartesian coordinate system. The parabola opens upwards, has its vertex in the fourth quadrant, and intersects the x-axis at (0,0) and approximately (7,0).

37.

Vertex: $(\frac{7}{2}, - \frac{37}{4})$ , axis of symmetry: $x = \frac{7}{2}$ , y-intercept: $(0, 3)$ , x-intercepts: $(\frac{7 + \sqrt{37}}{2}, 0), (\frac{7 - \sqrt{37}}{2}, 0)$

39.

Vertex: $(\frac{3}{2}, - 12)$ , axis of symmetry: $x = \frac{3}{2}$ , intercept: $(\frac{3 + 2 \sqrt{3}}{2}, 0)$ and $(\frac{3 - 2 \sqrt{3}}{2}, 0)$

A graph displays a blue parabola opening upwards. The parabola's vertex is in the fourth quadrant, intersecting the x-axis around (3.5, 0) and the y-axis around (0, -3).

41.

$f (x) = x^{2} + 2 x + 3$

43.

$f (x) = - 3 x^{2} - 6 x - 1$

45.

$f (x) = - \frac{1}{4} x^{2} - x + 2$

47.

$f (x) = x^{2} + 2 x + 1$

49.

$f (x) = - x^{2} + 2 x$

50.

$f (x) = 2 x^{2}$

53.

The graph is shifted to the right or left (a horizontal shift).

55.

The suspension bridge has 1,000 feet distance from the center.

57.

Domain is $(- \infty, \infty) .$ Range is $(- \infty, 2] .$

59.

Domain: $(- \infty, \infty)$ ; range: $[100, \infty)$

61.

$f (x) = 2 x^{2} + 2$

63.

$f (x) = - x^{2} - 2$

65.

$f (x) = 3 x^{2} + 6 x - 15$

67.

75 feet by 50 feet

69.

3 and 3; product is 9

71.

The revenue reaches the maximum value when 1800 thousand phones are produced.

73.

2.449 seconds

75.

41 trees per acre

5.2 Section Exercises

The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

As $x$ decreases without bound, so does $f (x) .$ As $x$ increases without bound, so does $f (x) .$

The polynomial function is of even degree and leading coefficient is negative.

Power function

Neither

11.

Neither

13.

Degree = 2, Coefficient = –2

15.

Degree =4, Coefficient = –2

17.

As $x \to \infty$ , $f (x) \to \infty, as x \to - \infty, f (x) \to \infty$

19.

As $x \to - \infty$ , $f (x) \to - \infty, as x \to \infty, f (x) \to - \infty$

21.

As $x \to - \infty$ , $f (x) \to - \infty, as x \to \infty, f (x) \to - \infty$

23.

As $x \to \infty$ , $f (x) \to \infty, as x \to - \infty, f (x) \to - \infty$

25.

y-intercept is $(0, 12),$ t-intercepts are $(1, 0); (- 2, 0); and (3, 0) .$

27.

y-intercept is $(0, - 16) .$ x-intercepts are $(2, 0)$ and $(- 2, 0) .$

29.

y-intercept is $(0, 0) .$ x-intercepts are $(0, 0), (4, 0),$ and $(- 2, 0) .$

31.

33.

35.

37.

39.

Yes. Number of turning points is 2. Least possible degree is 3.

41.

Yes. Number of turning points is 1. Least possible degree is 2.

43.

Yes. Number of turning points is 0. Least possible degree is 1.

45.

Yes. Number of turning points is 0. Least possible degree is 1.

47.

$x$	$f (x)$
10	9,500
100	99,950,000
–10	9,500
–100	99,950,000

As $x \to - \infty$ , $f (x) \to \infty, as x \to \infty, f (x) \to \infty$

49.

$x$	$f (x)$
10	–504
100	–941,094
–10	1,716
–100	1,061,106

As $x \to - \infty$ , $f (x) \to \infty, as x \to \infty, f (x) \to - \infty$

51.

The $y -$ intercept is $(0, 0) .$ The $x -$ intercepts are $(0, 0), (2, 0) .$ As $x \to - \infty$ , $f (x) \to \infty, as x \to \infty, f (x) \to \infty$

53.

The $y -$ intercept is $(0, 0)$ . The $x -$ intercepts are $(0, 0), (5, 0), (7, 0) .$ As $x \to - \infty$ , $f (x) \to - \infty, as x \to \infty, f (x) \to \infty$

55.

A graph on a coordinate plane displays a cubic function. The x-axis ranges from -10 to 10, and the y-axis ranges from -500 to 500. The blue curve illustrates the function, starting from approximately (-7, -400), increasing to a local maximum near (-1, 50), then decreasing to a local minimum near (3, -50), and finally increasing again, passing through (0, 0), (2, 0) and approximately (7, 300).

The $y -$ intercept is $(0, 0) .$ The $x -$ intercept is $(- 4, 0), (0, 0), (4, 0) .$ $A s x \to - \infty$ , $f (x) \to - \infty, as x \to \infty, f (x) \to \infty$

57.

The $y -$ intercept is $(0, - 81) .$ The $x -$ intercept are $(3, 0), (- 3, 0) .$ As $x \to - \infty$ , $f (x) \to \infty, as x \to \infty, f (x) \to \infty$

59.

The $y -$ intercept is $(0, 0) .$ The $x -$ intercepts are $(- 3, 0), (0, 0), (5, 0) .$ As $x \to - \infty$ , $f (x) \to - \infty, as x \to \infty, f (x) \to \infty$

61.

$f (x) = x^{2} - 4$

63.

$f (x) = x^{3} - 4 x^{2} + 4 x$

65.

$f (x) = x^{4} + 1$

67.

$V (m) = 8 m^{3} + 36 m^{2} + 54 m + 27$

69.

$V (x) = 4 x^{3} - 32 x^{2} + 64 x$

5.3 Section Exercises

The $x -$ intercept is where the graph of the function crosses the $x -$ axis, and the zero of the function is the input value for which $f (x) = 0.$

If we evaluate the function at $a$ and at $b$ and the sign of the function value changes, then we know a zero exists between $a$ and $b .$

There will be a factor raised to an even power.

$(- 2, 0), (3, 0), (- 5, 0)$

$(3, 0), (- 1, 0), (0, 0)$

11.

$(0, 0), (- 5, 0), (2, 0)$

13.

$(0, 0), (- 5, 0), (4, 0)$

15.

$(2, 0), (- 2, 0), (- 1, 0)$

17.

$(- 2, 0), (2, 0), (\frac{1}{2}, 0)$

19.

$(1, 0), (- 1, 0)$

21.

$(0, 0), (\sqrt{3}, 0), (- \sqrt{3}, 0)$

23.

$(0, 0), (1, 0), (- 1, 0), (2, 0), (- 2, 0)$

25.

$f (2) = - 10$ and $f (4) = 28.$ Sign change confirms.

27.

$f (1) = 3$ and $f (3) = - 77.$ Sign change confirms.

29.

$f (0.01) = 1.000001$ and $f (0.1) = - 7.999.$ Sign change confirms.

31.

0 with multiplicity 2, $- \frac{3}{2}$ with multiplicity 5, 4 with multiplicity 2

33.

0 with multiplicity 2, –2 with multiplicity 2

35.

$- \frac{2}{3}$ with multiplicity 5, 5 with multiplicity 2

37.

0 with multiplicity 4, 2 with multiplicity 1, −1 with multiplicity 1

39.

$\frac{3}{2}$ with multiplicity 2, 0 with multiplicity 3

41.

$0 with multiplicity 6, \frac{2}{3} with multiplicity 2$

43.

x-intercepts, $(1, 0)$ with multiplicity 2, $(- 4, 0)$ with multiplicity 1, $y -$ intercept $(0, 4).$ As $x \to - \infty, f (x) \to - \infty, as x \to \infty, f (x) \to \infty .$

45.

x-intercepts $(3, 0)$ with multiplicity 3, $(2, 0)$ with multiplicity 2, $y -$ intercept $(0, - 108) .$ As $x \to - \infty, f (x) \to - \infty, as x \to \infty, f (x) \to \infty .$

47.

x-intercepts $(0, 0), (- 2, 0), (4, 0)$ with multiplicity 1, $y -$ intercept $(0, 0) .$ As $x \to - \infty, f (x) \to \infty, as x \to \infty, f (x) \to - \infty .$

49.

$f (x) = - \frac{2}{9} (x - 3) (x + 1) (x + 3)$

51.

$f (x) = \frac{1}{4} {(x + 2)}^{2} (x - 3)$

53.

–4, –2, 1, 3 with multiplicity 1

55.

–2, 3 each with multiplicity 2

57.

$f (x) = - \frac{2}{3} (x + 2) (x - 1) (x - 3)$

59.

$f (x) = \frac{1}{3} {(x - 3)}^{2} {(x - 1)}^{2} (x + 3)$

61.

$f (x) = −15 {(x - 1)}^{2} {(x - 3)}^{3}$

63.

$f (x) = - 2 (x + 3) (x + 2) (x - 1)$

65.

$f (x) = - \frac{3}{2} {(2 x - 1)}^{2} (x - 6) (x + 2)$

67.

local max $(- .58, – .62),$ local min $(.58, –1 .38)$

69.

global min $(- .63, – .47)$

71.

global min $(.75, .89)$

73.

$f (x) = {(x - 500)}^{2} (x + 200)$

75.

$f (x) = 4 x^{3} - 36 x^{2} + 80 x$

77.

$f (x) = 4 x^{3} - 36 x^{2} + 60 x + 100$

79.

$f (x) = 9 π (x^{3} + 5 x^{2} + 8 x + 4)$

5.4 Section Exercises

The binomial is a factor of the polynomial.

$x + 6 + \frac{5}{x - 1}, quotient: x + 6, remainder: 5$

$3 x + 2, quotient: 3 x + 2, remainder: 0$

$x - 5, quotient: x - 5, remainder: 0$

$2 x - 7 + \frac{16}{x + 2}, quotient: 2 x - 7, remainder: 16$

11.

$x - 2 + \frac{6}{3 x + 1}, quotient: x - 2, remainder: 6$

13.

$2 x^{2} - 3 x + 5, quotient: 2 x^{2} - 3 x + 5, remainder: 0$

15.

$2 x^{2} + 2 x + 1 + \frac{10}{x - 4}$

17.

$2 x^{2} - 7 x + 1 - \frac{2}{2 x + 1}$

19.

$3 x^{2} - 11 x + 34 - \frac{106}{x + 3}$

21.

$x^{2} + 5 x + 1$

23.

$4 x^{2} - 21 x + 84 - \frac{323}{x + 4}$

25.

$x^{2} - 14 x + 49$

27.

$3 x^{2} + x + \frac{2}{3 x - 1}$

29.

$x^{3} - 3 x + 1$

31.

$x^{3} - x^{2} + 2$

33.

$x^{3} - 6 x^{2} + 12 x - 8$

35.

$x^{3} - 9 x^{2} + 27 x - 27$

37.

$2 x^{3} - 2 x + 2$

39.

Yes $(x - 2) (3 x^{3} - 5)$

41.

Yes $(x - 2) (4 x^{3} + 8 x^{2} + x + 2)$

43.

45.

$(x - 1) (x^{2} + 2 x + 4)$

47.

$(x - 5) (x^{2} + x + 1)$

49.

$Quotient: 4 x^{2} + 8 x + 16, remainder: - 1$

51.

$Quotient: 3 x^{2} + 3 x + 5, remainder: 0$

53.

$Quotient: x^{3} - 2 x^{2} + 4 x - 8, remainder: - 6$

55.

$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$

57.

$x^{3} - x^{2} + x - 1 + \frac{1}{x + 1}$

59.

$1 + \frac{1 + i}{x - i}$

61.

$1 + \frac{1 - i}{x + i}$

63.

$x^{2} + i x - 1 + \frac{1 - i}{x - i}$

65.

$2 x^{2} + 3$

67.

$2 x + 3$

69.

$x + 2$

71.

$x - 3$

73.

$3 x^{2} - 2$

5.5 Section Exercises

The theorem can be used to evaluate a polynomial.

Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

$- 106$

$0$

11.

$255$

13.

$- 1$

15.

$- 2, 1, \frac{1}{2}$

17.

$- 2$

19.

$- 3$

21.

$- \frac{5}{2}, \sqrt{6}, - \sqrt{6}$

23.

$2, - 4, - \frac{3}{2}$

25.

$4, - 4, - 5$

27.

$5, - 3, - \frac{1}{2}$

29.

$\frac{1}{2}, \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

31.

$\frac{3}{2}$

33.

$2, 3, - 1, - 2$

35.

$\frac{1}{2}, - \frac{1}{2}, 2, - 3$

37.

$- 1, - 1, \sqrt{5}, - \sqrt{5}$

39.

$- \frac{3}{4}, - \frac{1}{2}$

41.

$2, 3 + 2 i, 3 - 2 i$

43.

$- \frac{2}{3}, 1 + 2 i, 1 - 2 i$

45.

$- \frac{1}{2}, 1 + 4 i, 1 - 4 i$

47.

1 positive, 1 negative

49.

3 or 1 positive, 0 negative

51.

0 positive, 3 or 1 negative

53.

2 or 0 positive, 2 or 0 negative

55.

2 or 0 positive, 2 or 0 negative

57.

$\pm 5, \pm 1, \pm \frac{5}{2}, \pm \frac{1}{2}$

59.

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}$

61.

$1, \frac{1}{2}, - \frac{1}{3}$

63.

$2, \frac{1}{4}, - \frac{3}{2}$

65.

$\frac{5}{4}$

67.

$f (x) = \frac{4}{9} (x^{3} + x^{2} - x - 1)$

69.

$f (x) = - \frac{1}{5} (4 x^{3} - x)$

71.

8 by 4 by 6 inches

73.

5.5 by 4.5 by 3.5 inches

75.

8 by 5 by 3 inches

77.

Radius = 6 meters, Height = 2 meters

79.

Radius = 2.5 meters, Height = 4.5 meters

5.6 Section Exercises

The rational function will be represented by a quotient of polynomial functions.

The numerator and denominator must have a common factor.

Yes. The numerator of the formula of the functions would have only complex roots and/or factors common to both the numerator and denominator.

$All reals x \neq - 1, 1$

$All reals x \neq - 1, - 2, 1, 2$

11.

V.A. at $x = - \frac{2}{5};$ H.A. at $y = 0;$ Domain is all reals $x \neq - \frac{2}{5}$

13.

V.A. at $x = 4, - 9;$ H.A. at $y = 0;$ Domain is all reals $x \neq 4, - 9$

15.

V.A. at $x = 0, 4, - 4;$ H.A. at $y = 0;$ Domain is all reals $x \neq 0, 4, - 4$

17.

V.A. at $x = 5;$ H.A. at $y = 0;$ Domain is all reals $x \neq 5, - 5$

19.

V.A. at $x = \frac{1}{3};$ H.A. at $y = - \frac{2}{3};$ Domain is all reals $x \neq \frac{1}{3} .$

21.

none

23.

$x -intercepts none, y -intercept (0, \frac{1}{4})$

25.

Local behavior: $x \to - {\frac{1}{2}}^{+}, f (x) \to - \infty, x \to - {\frac{1}{2}}^{-}, f (x) \to \infty$

End behavior: $x \to \pm \infty, f (x) \to \frac{1}{2}$

27.

Local behavior: $x \to 6^{+}, f (x) \to - \infty, x \to 6^{-}, f (x) \to \infty,$ End behavior: $x \to \pm \infty, f (x) \to - 2$

29.

Local behavior: $x \to {\frac{1}{3}}^{+}, f (x) \to - \infty, x \to {\frac{1}{3}}^{-},$ $f (x) \to \infty, x \to {- \frac{5}{2}}^{+}, f (x) \to \infty, x \to {- \frac{5}{2}}^{-}, f (x) \to - \infty$

End behavior: $x \to \pm \infty, f (x) \to \frac{1}{3}$

31.

$y = 2 x + 4$

33.

$y = 2 x$

35.

$V . A . x = 0, H . A . y = 2$

37.

$V . A . x = 2, H . A . y = 0$

39.

$V . A . x = - 4, H . A . y = 2; (\frac{3}{2}, 0); (0, - \frac{3}{4})$

Graph of p(x)=(2x-3)/(x+4) with its vertical asymptote at x=-4 and horizontal asymptote at y=2.

41.

$V . A . x = 2, H . A . y = 0, (0, 1)$

Graph of s(x)=4/(x-2)^2 with its vertical asymptote at x=2 and horizontal asymptote at y=0.

43.

$V . A . x = - 4, x = \frac{4}{3}, H . A . y = 1; (5, 0); (- \frac{1}{3}, 0); (0, \frac{5}{16})$

Graph of f(x)=(3x^2-14x-5)/(3x^2+8x-16) with its vertical asymptotes at x=-4 and x=4/3 and horizontal asymptote at y=1.

45.

$V . A . x = - 1, H . A . y = 1; (- 3, 0); (0, 3)$

Graph of a(x)=(x^2+2x-3)/(x^2-1) with its vertical asymptote at x=-1 and horizontal asymptote at y=1.

47.

$V . A . x = 4, S . A . y = 2 x + 9; (- 1, 0); (\frac{1}{2}, 0); (0, \frac{1}{4})$

Graph of h(x)=(2x^2+x-1)/(x-1) with its vertical asymptote at x=4 and slant asymptote at y=2x+9.

49.

$V . A . x = - 2, x = 4, H . A . y = 1, (1, 0); (5, 0); (- 3, 0); (0, - \frac{15}{16})$

Graph of w(x)=(x-1)(x+3)(x-5)/(x+2)^2(x-4) with its vertical asymptotes at x=-2 and x=4 and horizontal asymptote at y=1.

51.

$y = 50 \frac{x^{2} - x - 2}{x^{2} - 25}$

53.

$y = 7 \frac{x^{2} + 2 x - 24}{x^{2} + 9 x + 20}$

55.

$y = \frac{1}{2} \frac{x^{2} - 4 x + 4}{x + 1}$

57.

$y = 4 \frac{x - 3}{x^{2} - x - 12}$

59.

$y = \frac{27 (x - 2)}{(x + 3) {(x - 3)}^{2}}$

61.

$y = \frac{1}{3} \frac{x^{2} + x - 6}{x - 1}$

63.

$y = - 6 \frac{{(x - 1)}^{2}}{(x + 3) {(x - 2)}^{2}}$

65.

$x$	2.01	2.001	2.0001	1.99	1.999
$y$	100	1,000	10,000	–100	–1,000

$x$	10	100	1,000	10,000	100,000
$y$	.125	.0102	.001	.0001	.00001

Vertical asymptote $x = 2,$ Horizontal asymptote $y = 0$

67.

$x$	–4.1	–4.01	–4.001	–3.99	–3.999
$y$	82	802	8,002	–798	–7998

$x$	10	100	1,000	10,000	100,000
$y$	1.4286	1.9331	1.992	1.9992	1.999992

Vertical asymptote $x = - 4,$ Horizontal asymptote $y = 2$

69.

$x$	–.9	–.99	–.999	–1.1	–1.01
$y$	81	9,801	998,001	121	10,201

$x$	10	100	1,000	10,000	100,000
$y$	.82645	.9803	.998	.9998

Vertical asymptote $x = - 1,$ Horizontal asymptote $y = 1$

71.

$(\frac{3}{2}, \infty)$

73.

$(- 2, 1) \cup (4, \infty)$

75.

$(2, 4)$

77.

$(2, 5)$

79.

$(- 1, 1)$

81.

$C (t) = \frac{8 + 2 t}{300 + 20 t}$

83.

After about 6.12 hours.

85.

$A (x) = 50 x^{2} + \frac{800}{x} .$ 2 by 2 by 5 feet.

87.

$A (x) = π x^{2} + \frac{100}{x} .$ Radius = 2.52 meters.

5.7 Section Exercises

It can be too difficult or impossible to solve for $x$ in terms of $y .$

We will need a restriction on the domain of the answer.

$f^{- 1} (x) = \sqrt{x} + 4$

$f^{- 1} (x) = \sqrt{x + 3} - 1$

$f^{- 1} (x) = \sqrt{12 - x}$

11.

$f^{- 1} (x) = \sqrt{\frac{x - 4}{2}}$

13.

$f^{- 1} (x) = \sqrt{\frac{x - 1}{3} 3}$

15.

$f^{- 1} (x) = \sqrt{\frac{4 - x}{2} 3}$

17.

$f^{−1} (x) = \frac{3 - x^{2}}{4}, [0, \infty)$

19.

$f^{−1} (x) = \frac{{(x - 5)}^{2} + 8}{6}$

21.

$f^{−1} (x) = {(3 - x)}^{2}$

23.

$f^{−1} (x) = \frac{4 x + 3}{x}$

25.

$f^{−1} (x) = \frac{7 x - 3}{1 - x}$

27.

$f^{−1} (x) = \frac{2 x - 1}{5 x + 5}$

29.

$f^{−1} (x) = \sqrt{x + 3} - 2$

31.

A graph displays two curved lines on a coordinate plane with x and y axes ranging from -2 to 8. A blue curve starts near (0.5, 2) and curves upwards and to the right, appearing to be an exponential function. An orange curve starts near (2, 0.5) and curves to the right and slightly upwards, appearing to be a logarithmic function. The curves suggest an inverse relationship.

$f^{- 1} (x) = \sqrt{x - 2}$

33.

Two inverse functions plotted on a coordinate plane: an exponential curve (orange) in the second and first quadrants, and a logarithmic curve (blue) in the first and fourth quadrants.

$f^{- 1} (x) = \sqrt{x - 3}$

35.

A graph displaying a blue curve and an orange curve on an x-y coordinate plane. The blue curve passes through (0,2) and (1,6), while the orange curve passes through (2,0) and (6,1). These curves are inverses of each other, reflected across the y=x line.

$f^{- 1} (x) = \sqrt[3]{x - 3}$

37.

A Cartesian graph displays two distinct increasing functions: a blue curve with an upward concave shape and an orange curve with a downward concave shape.

$f^{- 1} (x) = \sqrt{x + 4} - 2$

39.

The graph of the reciprocal function f(x)=1/x, showing two branches in the first and third quadrants. There is a vertical asymptote at x=0 (the y-axis) and a horizontal asymptote at y=0 (the x-axis).

41.

A Cartesian coordinate system displays a graph with two separate curves. The first curve starts at approximately (-1, 0) and extends upwards as it approaches x=0 from the left, showing a steep increase. The second curve starts at approximately (1, 0) and extends towards the upper right, gradually increasing as x increases. The x-axis is labeled from -4 to 4, and the y-axis from -4 to 4, with grid lines at integer values.

$[- 1, 0) \cup [1, \infty)$

43.

A graph displays a curve with a vertical asymptote at x=4. The curve descends from high y-values for x > 4, then gradually flattens out to the right, approaching a horizontal asymptote.

$[- 3, 0] \cup (4, \infty)$

45.

The graph displays a function f(x) on a Cartesian coordinate system. The x-axis and y-axis both range from -32 to 32, with grid lines every 8 units. The function exhibits a vertical asymptote near x = -4. From the left of this asymptote, the function approaches a value around y=6, then curves sharply downwards towards negative infinity as x decreases. From the right of x = -4, the function shoots upwards towards positive infinity. There appears to be another vertical asymptote near x = 1. As x approaches 1 from the left, the function goes towards negative infinity, while from the right, it rises towards positive infinity. Between these two asymptotes, the function forms a small curve with a local maximum around (0, 1). As x approaches negative infinity, the function seems to level off, approaching a horizontal asymptote around y = 8.

$[- ∞, - 4] \cdot [- 3, 3]$

47.

A coordinate plane shows the graph of a cubic function. The curve passes through the origin (0,0) and extends from the third quadrant, through (1,1), into the first quadrant, showing increasing values as x increases.

$(- 2, 0), (0, 1), (8, 2)$

49.

A graph showing a cubic function plotted on a coordinate plane. The blue curve passes through the origin (0,0) and increases from approximately (-2, -20) to (2, 20).

$(- 13, - 1), (- 4, 0), (5, 1)$

51.

$f^{- 1} (x) = \sqrt[3]{\frac{x - b}{a}}$

53.

$f^{- 1} (x) = \frac{\sqrt{x^{2} - b}}{a}$

55.

$f^{- 1} (x) = \frac{c x - b}{a - x}$

57.

$t (h) = \sqrt{\frac{600 - h}{16}}$ , 3.54 seconds

59.

$r (A) = \sqrt{\frac{A}{4 π}}, \approx$ 8.92 in.

61.

$l (T) = 32.2 (\frac{T}{2 π}), \approx$ 3.26 ft

63.

$r (A) = \sqrt{\frac{A + 8 π}{2 π}}$ –2, 3.99 ft

65.

$r (V) = \sqrt{\frac{V}{10 π}},$ ≈ 5.64 ft

5.8 Section Exercises

The graph will have the appearance of a power function.

No. Multiple variables may jointly vary.

$y = 5 x^{2}$

$y = \frac{1}{1944} x^{3}$

$y = 6 x^{4}$

11.

$y = \frac{18}{x^{2}}$

13.

$y = \frac{81}{x^{4}}$

15.

$y = \frac{20}{\sqrt[3]{x}}$

17.

$y = 10 x z w$

19.

$y = 10 x \sqrt{z}$

21.

$y = 4 \frac{x z}{w}$

23.

$y = 40 \frac{x z}{\sqrt{w} t^{2}}$

25.

$y = 256$

27.

$y = 6$

29.

$y = 6$

31.

$y = 27$

33.

$y = 3$

35.

$y = 18$

37.

$y = 90$

39.

$y = \frac{81}{2}$

41.

$y = \frac{3}{4} x^{2}$

43.

$y = \frac{1}{3} \sqrt{x}$

45.

$y = \frac{4}{x^{2}}$

A graph of a function with a vertical dashed asymptote at x=0 and a horizontal dashed asymptote at y=0. The blue curve is symmetric about the y-axis, approaching infinity near x=0 and zero as x extends horizontally.

47.

1.89 years

49.

0.61 years

51.

3 seconds

53.

48 inches

55.

49.75 pounds

57.

33.33 amperes

59.

2.88 inches

Review Exercises

$f (x) = {(x - 2)}^{2} - 9 vertex (2, –9), intercepts (5, 0); (–1, 0); (0, –5)$

$f (x) = \frac{3}{25} {(x + 2)}^{2} + 3$

300 meters by 150 meters, the longer side parallel to river.

Yes, degree = 5, leading coefficient = 4

Yes, degree = 4, leading coefficient = 1

11.

$As x \to - \infty, f (x) \to - \infty, as x \to \infty, f (x) \to \infty$

13.

–3 with multiplicity 2, $\frac{1}{2}$ with multiplicity 1, –1 with multiplicity 3

15.

4 with multiplicity 1

17.

$\frac{1}{2}$ with multiplicity 1, 3 with multiplicity 3

19.

$x^{2} + 4$ with remainder 12

21.

$x^{2} - 5 x + 20 - \frac{61}{x + 3}$

23.

$2 x^{2} - 2 x - 3$ , so factored form is $(x + 4) (2 x^{2} - 2 x - 3)$

25.

${- 2, 4, - \frac{1}{2}}$

27.

${1, 3, 4, \frac{1}{2}}$

29.

0 or 2 positive, 1 negative

31.

Intercepts $(–2, 0) and (0, - \frac{2}{5})$ , Asymptotes $x = 5$ and $y = 1.$

33.

Intercepts (3, 0), (-3, 0), and $(0, \frac{27}{2})$ , Asymptotes $x = 1, x = - 2, y = 3.$

35.

$y = x - 2$

37.

$f^{- 1} (x) = \sqrt{x} + 2$

39.

$f^{- 1} (x) = \sqrt{x + 11} - 3$

41.

$f^{- 1} (x) = \frac{{(x + 3)}^{2} - 5}{4}, x \geq - 3$

43.

$y = 64$

45.

$y = 72$

47.

148.5 pounds

Practice Test

Degree: 5, leading coefficient: −2

$As x \to −∞, f (x) \to ∞, As x \to ∞, f (x) \to ∞$

$f (x) = 3 {(x - 2)}^{2}$

3 with multiplicity 3, $\frac{1}{3}$ with multiplicity 1, 1 with multiplicity 2

$- \frac{1}{2}$ with multiplicity 3, 2 with multiplicity 2

11.

$x^{3} + 2 x^{2} + 7 x + 14 + \frac{24}{x - 2}$

13.

${–3, –1, \frac{3}{2}}$

15.

1, −2, and − $\frac{3}{2}$ (multiplicity 2)

17.

$f (x) = - \frac{2}{3} {(x - 3)}^{2} (x - 1) (x + 2)$

19.

2 or 0 positive, 1 negative

21.

$(- 3, 0) (1, 0) (0, \frac{3}{4})$

A graph displays a rational function with three distinct branches. The x-axis ranges from -10 to 10, and the y-axis ranges from -8 to 8, both with increments of 2. There are two vertical dashed asymptotes, one labeled "x = -2" in orange and the other labeled "x = 2" in teal. There is also a horizontal dashed asymptote labeled "y = 1" in red. The function's graph approaches these asymptotes. The left branch approaches x = -2 from the left, crosses the x-axis at approximately -5, and then approaches y = 1 from below as x goes to negative infinity. The middle branch is located between x = -2 and x = 2. It starts from positive infinity near x = -2, decreases, crosses the y-axis at around (0, 0.25), and then goes down to negative infinity near x = 2. The right branch starts from positive infinity near x = 2, decreases, approaches y = 1 from above as x goes to positive infinity.

23.

$f^{- 1} (x) = {(x - 4)}^{2} + 2, x \geq 4$

25.

$f^{- 1} (x) = \frac{x + 3}{3 x - 2}$

27.

$y = 20$