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Defining Functions by a Rule

4.4.2 Defining Functions by a Rule

4.4.2 • Defining Functions by a Rule

Activity

For questions 1 – 4, match each equation with a verbal description that represents the same function.

1. f ( x ) = 3 x − 7 f ( x ) = 3 x − 7

  1. Subtract 7 from the input, then divide the result by 3.

  2. Subtract 7 from the input, then multiply the result by 3.

  3. Multiply the input by 3, then subtract 7 from the result.

  4. Divide the input by 3, and then subtract 7 from the result.

Multiply the input by 3, then subtract 7 from the result.

2. g ( x ) = 3 ( x − 7 ) g ( x ) = 3 ( x − 7 )

  1. Subtract 7 from the input, then divide the result by 3.

  2. Subtract 7 from the input, then multiply the result by 3.

  3. Multiply the input by 3, then subtract 7 from the result.

  4. Divide the input by 3, and then subtract 7 from the result.

Multiply the input by 3, then subtract 7 from the result.

3. h ( x ) = x 3 − 7 h ( x ) = x 3 − 7

  1. Subtract 7 from the input, then divide the result by 3.

  2. Subtract 7 from the input, then multiply the result by 3.

  3. Multiply the input by 3, then subtract 7 from the result.

  4. Divide the input by 3, and then subtract 7 from the result.

Divide the input by 3, and then subtract 7 from the result.

4. h ( x ) = x − 7 3 h ( x ) = x − 7 3

  1. Subtract 7 from the input, then divide the result by 3.

  2. Subtract 7 from the input, then multiply the result by 3.

  3. Multiply the input by 3, then subtract 7 from the result.

  4. Divide the input by 3, and then subtract 7 from the result.

Subtract 7 from the input, then divide the result by 3.

5. For one of the functions, when the input is 6, the output is –3. Which is that function: f f , g g , h h , or k k ?

g g

6. Explain how you knew the answer to question 5.

Compare your answers: Your answers may vary.

k ( x ) k ( x ) . When x is 0, k ( x ) k ( x ) has the greatest value because k ( 0 ) = 0 − 7 3 = − 7 3 = − 2 1 3 k ( 0 ) = 0 − 7 3 = − 7 3 = − 2 1 3 .

7. Which function value, f f , g g , h h , or k k is the greatest when the input is 0.

When x x is 0, k ( x ) k ( x ) has the greatest value.

8. Which function value, f f , g g , h h , or k k is the greatest when the input is 10.

f ( x ) f ( x ) . When x is 10, f ( 10 ) = 3 ( 10 ) − 7 = 30 − 7 = 23 f ( 10 ) = 3 ( 10 ) − 7 = 30 − 7 = 23 and, 23 is larger than g ( 10 ) = 9 , h ( 10 ) = − 11 3 g ( 10 ) = 9 , h ( 10 ) = − 11 3 , and k ( 10 ) = 1 k ( 10 ) = 1

Are you ready for more?

Extending Your Thinking

Mai says f ( x ) f ( x ) is always greater than g ( x ) g ( x ) for the same value of x x . Is this true? Explain how you know.

Compare your answer:

Your answer may vary, but here is a sample. Mai's statement is true. We can start by rewriting g ( x ) g ( x ) as g ( x ) = 3 x − 21 g ( x ) = 3 x − 21 . Then, to get the output of both functions, we first multiply the input, x x , by 3 to get 3 x 3 x , and then we subtract a value from it. For function f f , we subtract 7. For function g g , we subtract 21. The output of g g will always be 14 less than that of f f .

Self Check

Which function has the least value when x = 0 ?
  1. f ( x ) = − 4 x + 5
  2. f ( x ) = 2 ( x − 1 )
  3. f ( x ) = 3 x
  4. f ( x ) = 2 x + 1

Additional Resources

Evaluating Functions with Function Notation

Find the Value of a Function

It is very convenient to name a function, and most often we name it f f , g g , or h h . In any function, for each x x -value from the domain we get a corresponding y y -value in the range. For the function f f , we write this range value y y as f ( x ) f ( x ) . This is called function notation, and is read f f of x x or the value of f f at x x . In this case, the parentheses do not indicate multiplication.

Function Notation

For the function y = f ( x ) y = f ( x ) :

  • f f is the name of the function.
  • x x is the input value.
  • f ( x ) f ( x ) is the output value, y y , corresponding to the value of x x .

We read f ( x ) f ( x ) as f f of x x or the value of f f at x x .

We call x x the independent variable as it can be any value in the domain. We call y y the dependent variable as its value depends on x x .

Independent and Dependent Variables

For the function y = f ( x ) y = f ( x ) ,

x x is the independent variable as it can be any value.

y y is the dependent variable as its value depends on x x .

Much as when you first encountered the variable x x , function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.

Let’s look at the equation y = 4 x − 5 y = 4 x − 5 . To find the value of y y when x = 2 x = 2 , we know to substitute x = 2 x = 2 into the equation and then simplify.

Original Equation y = 4 x − 5 y = 4 x − 5

Substitute x = 2 x = 2 in for x x .

y = 4 ( 2 ) − 5 y = 4 ( 2 ) − 5

Simplify.

y = 3 y = 3

The value of the function at x = 2 x = 2 is 3.

We can do the same thing using function notation. The equation y = 4 x − 5 y = 4 x − 5 can be written as f ( x ) = 4 x − 5 f ( x ) = 4 x − 5 . To find the value when x = 2 x = 2 , we write:

Original Equation f ( x ) = 4 x − 5 f ( x ) = 4 x − 5
Substitute x = 2 x = 2 in for x x . f ( 2 ) = 4 ( 2 ) − 5 f ( 2 ) = 4 ( 2 ) − 5
Simplify. f ( 2 ) = 3 f ( 2 ) = 3

The value of the function at x = 2 x = 2 is 3.

This process of finding the value of f ( x ) f ( x ) for a given value of x x is called evaluating the function.

Try it

Evaluating Functions with Function Notation

Find the value of f ( x ) = 5 x + 2 f ( x ) = 5 x + 2 when x = 3 x = 3 .

Here is how to evaluate a function:

Original Equation

f ( x ) = 5 x + 2 f ( x ) = 5 x + 2

Substitute x = 3 x = 3 in for x x .

f ( 3 ) = 5 ( 3 ) + 2 f ( 3 ) = 5 ( 3 ) + 2

Simplify.

f ( 3 ) = 17 f ( 3 ) = 17

Steps to evaluate the function f(x) = 5x + 2 at x = 3.