Evaluate a Function: Answers and Explanations

The notation, ƒ(x), introduces a function. Use this article to answer the following questions.

• What are some examples of ƒ(x)?
• What do these examples mean?
• How is ƒ(x) used in the real world?
• What are the steps to finding a specific value of a function?

Examples of ƒ(x) Decoded

• ƒ(x) = x + 3   (The value of x determines ƒ(x).)
• ƒ(t) = 3t³   (The value of t determines ƒ(t).)
• ƒ(s) = -100s² + 5s     (The value of s determines ƒ(s).)

Real World Examples of ƒ(x)

The real world — round, fast-paced, expensive — relies on functions. 

• The circumference of a circle, C(r), depends on its radius, r.  C(r) = 2Πr
  
• The area of a circle, A(r), depends on its radius, r.  A(r) = Πr²
• The distance, D(r), from home to work depends on the time, t, spent driving at an average speed of 50 miles an hour. D(r) = 50t
  
• The value, V(t), of a $500 investment, with an annual return of 4%, depends on t years. V(t) = 500(1.04)^t
  

Finding Specific Values of Functions

Use these examples to learn how to evaluate a function, or to find the specific value of ƒ(x) at a certain point.

»»»Example 1

ƒ(x) = x + 3; Find ƒ(3)

1. ƒ(3) = 3 + 3 (Replace x with 3.)
2. ƒ(3) = 6       (Simplify.)

»»»Example 2

g(x) = 2x³ + 5; Find g(-4)

1. g(-4) = 2(-4)³ + 5       (Replace x with -4)
2. g(-4) = 2 * -64 + 5     (Simplify.)
3. g(-4) = -128 + 5         (Simplify.)
4. g(-4) = -123               (Simplify.)

»»»Example 3

h(x) = sinx; Find h(180°)

1. h(180°) = sin180°   (Replace x with 180°.)
2. h(180°) = 0             (Use the calculator to simplify.)

Answers and Explanations

1. ƒ(x) = 7x - 3; Find ƒ(0). ƒ(0) = -3

1. ƒ(0) = 7(0) - 3     (Replace x with 0.)
2. ƒ(0) = -3             (Simplify.)

2. ƒ(t) = |5t|; Find ƒ(2). ƒ(2) = 10

1. ƒ(2) = |5(2)|         (Replace t with 2.)
2. ƒ(2)  = 10             (Simplify.)

3. g(x) = x² + 8x - 6; Find g(1). g(1) = 3

1. g(1)= (1)² + 8(1) - 6     (Replace x with 0.)
2. g(1)= 1 + 8 - 6     (Simplify.)
3. g(1) = 3               (Simplify.)

4. ƒ(b) = 3^b; Find ƒ(3). ƒ(3) = 27

1. ƒ(3) = 3³     (Replace b with 0.)
2. ƒ(3) = 27   (Simplify.)

5. ƒ(x) = sinx; Find ƒ(45°). ƒ(45°) = .707

1. ƒ(45°) = sin45°     (Replace x with 45°.)
2. ƒ(45°) = .707     (Use the calculator to simplify.)

6. ƒ(x) = cosx; Find ƒ(90°). ƒ(90°) = 0

1. ƒ(90°) = cos90°     (Replace x with 90°.)
2. ƒ(90°) = 0             (Use the calculator to simplify.)

7. ƒ(c) = .25^c ; Find ƒ(-2). ƒ(-2) = 16

1. ƒ(-2) = .25^(-2)       (Replace c with -2.)
2. ƒ(-2) = 1/(.25)²    (Simplify.)
3. ƒ(-2) = 1/.0625   (Simplify.)
4. ƒ(-2) = 16          (Simplify.)

8. ƒ(x) = -10x + 5; Find ƒ(-10). ƒ(-10) = 105

1. ƒ(-10) = -10(-10) + 5    (Replace x with -10.)
2. ƒ(-10) = 100 + 5 = 105  (Simplify.)

9. ƒ(r) = 2Πr² + 2Πr; Find ƒ(4). ƒ(4) = 40Π

1. 2Π(4)² + 2Π(4)  (Replace r with 4.)
2. 2Π * 16 + 8Π    (Simplify.)
   
3. 32Π + 8Π = 40Π  (Simplify.)
   

10. ƒ(t) = 100(1.10)^t; Find ƒ(4). ƒ(4)= 146.41

1. ƒ(4) = 100(1.10)⁴      (Replace t with 4.)
2. ƒ(4) = 100*1.4641    (Simplify.)
   
3. ƒ(4) = 146.41           (Simplify.)