Comparing Graphs and Statements that Represent Functions without a Context

4.10.4 Comparing Graphs and Statements that Represent Functions without a Context

4.10.4 • Comparing Graphs and Statements that Represent Functions without a Context

Activity

Here are graphs that represent two functions, $f$ and $g$ .

A graph with two curves: f, a black upward then downward parabola peaking at y equals 8, and g, a blue straight line sloping downward from y equals 8. The x-axis ranges from 0 to 12, and the y-axis ranges from 0 to 9.

Decide which function value is greater for each given input. Be prepared to explain your reasoning.

1. Which is greater, $f (2)$ or $g (2)$

Compare your answer:

$g (2)$ is greater.

2. Which is greater, $f (4)$ or $g (4)$

Compare your answer:

They appear to be equal.

3. Which is greater, $f (6)$ or $g (6)$

Compare your answer:

$f (6)$ is greater.

4. Which is greater, $f (8)$ or $g (8)$

Compare your answer:

$f (8)$ is greater.

5. Is there a value of $x$ at which the equation $f (x) = g (x)$ is true? Be prepared to show your reasoning.

Compare your answers:

Yes. For example: If the graphs intersect when $x$ is 4 and when it is 9, then $f (4) = g (4)$ and $f (9) = g (9)$ .

6. Identify at least two values of $x$ at which the inequality $f (x) < g (x)$ is true.

Compare your answers:

1, 3, 10, 11, or other $x$ -values that are less than 4 and greater than 9

Self Check

Which of the following is true about the graphs of $f(x)$ and $g(x)$ below?

GRAPH OF TWO FUNCTIONS. ONE IS A PARABOLA THAT OPENS UP. THE OTHER IS A LINE WITH A NEGATIVE SLOPE. THE FUNCTIONS INTERSECT AT THE POINTS (1, −3) AND (−2, 0).

$f(-1)=g(-1)$
$f(-3)<g(-3)$
$f(1)=g(1)$
$f(0)>g(0)$

Additional Resources

Using Graphs to Find Values

Two functions, $f (x)$ and $g (x)$ , are graphed below.

A graph showing two functions: f(x), a parabola opening upward with a vertex of (negative 4,0) and x-intercepts at negative 2 and 2; and g(x), a straight downward-sloping line. The curves intersect near the points (negative 2, negative 2) and (1, negative 3) on the coordinate plane.

What is the value of $f (- 1)$ ?

Look at where $x = - 1$ touches the graph of $f$ . The $y$ -value is –3.

$f (- 1) = - 3$

What is the value of $g (- 1)$ ?

Look where $x = - 1$ touches the graph of $g$ . The $y$ -value is –1.

$g (- 1) = - 1$

Which value is greater?

$g (- 1) > f (- 1)$

Try it

Using Graphs to Find Values

Two functions, $f (x)$ and $g (x)$ , are graphed below.

Graph showing two functions: f of x, a downward-opening parabola, and g of x, a straight downward-sloping line. Both intersect the y-axis above 0. The axes are labeled, and the curves are named.

Which is greater, $f (1)$ or $g (1)$ ?

Here is how to determine which value is greater:

Looking at the graph, $f (1)$ is where $x = 1$ reaches the graph of $f (x)$ . The $y$ -value there is 4. So, $f (1) = 4$ .

Similarly, $g (1)$ is where $x = 1$ reaches the graph of $g (x)$ . The $y$ -value there is 2.

So, $f (1) > g (1)$ .