1. Introduction
So far we have seen a function f of a variable x represented by f(x). We have graphed f(x) and learned that its range is dependent on its domain. But, can a function be applied to expressions other than x? What would it mean if we wrote f(2x) or f(x + 1)? In this lesson, we will explore function notation and the versatility of functions.
3.1.4: Function Notation and Evaluating Functions

2. Introduction, continued
For example, let f be a function with the domain {1, 2, 3} and let f(x) = 2x. To evaluate f over the domain {1, 2, 3}, we would write the following equations by substituting each value in the domain for x:
f(1) = 2(1) = 2
f(2) = 2(2) = 4
f(3) = 2(3) = 6
{2, 4, 6} is the range of f(x).
3.1.4: Function Notation and Evaluating Functions

3. Key Concepts Functions can be evaluated at values and variables.
To evaluate a function, substitute the values for the domain for all occurrences of x.
To evaluate f(2) in f(x) = x + 1, replace all x’s with 2
and simplify: f(2) = (2) + 1 = 3. This means that f(2) = 3.
(x, (f(x)) is an ordered pair of a function and a point on the graph of the function.
3.1.4: Function Notation and Evaluating Functions

4. Common Errors/Misconceptions
thinking function notation means “f times x” instead of “f of x”
trying to multiply the left side of the function notation
3.1.4: Function Notation and Evaluating Functions

5. Guided Practice Example 1
Evaluate f(x) = 4x – 7 over the domain {1, 2, 3, 4}. What is the range?
3.1.4: Function Notation and Evaluating Functions

6. Guided Practice: Example 1, continued
To evaluate f(x) = 4x – 7 over the domain
{1, 2, 3, 4}, substitute the values from the domain into f(x) = 4x – 7.
3.1.4: Function Notation and Evaluating Functions

7. Guided Practice: Example 1, continued Evaluate f(1).
f(x) = 4x – 7
Original function
f(1) = 4(1) – 7
Substitute 1 for x.
f(1) = 4 – 7 = –3
Simplify.
3.1.4: Function Notation and Evaluating Functions

8. Guided Practice: Example 1, continued Evaluate f(2).
f(x) = 4x – 7
Original function
f(2) = 4(2) – 7
Substitute 2 for x.
f(2) = 8 – 7 = 1
Simplify.
3.1.4: Function Notation and Evaluating Functions

9. Guided Practice: Example 1, continued Evaluate f(3).
f(x) = 4x – 7
Original function
f(3) = 4(3) – 7
Substitute 3 for x.
f(3) = 12 – 7 = 5
Simplify.
3.1.4: Function Notation and Evaluating Functions

10. Guided Practice: Example 1, continued Evaluate f(4).
f(x) = 4x – 7
Original function
f(4) = 4(4) – 7
Substitute 4 for x.
f(4) = 16 – 7 = 9
Simplify.
3.1.4: Function Notation and Evaluating Functions

11. ✔ Guided Practice: Example 1, continued
Collect the set of outputs from the inputs.
The range is {–3, 1, 5, 9}. ✔
3.1.4: Function Notation and Evaluating Functions

12. Guided Practice: Example 1, continued 12
3.1.4: Function Notation and Evaluating Functions

13. Guided Practice Example 3
Raven started an online petition calling for more vegan options in the school cafeteria. So far, the number of signatures has doubled every day. She started with 32 signatures on the first day. Raven’s petition can be modeled by the function f(x) = 32(2)x. Evaluate f(3) and interpret the results in terms of the petition.
3.1.4: Function Notation and Evaluating Functions

14. Guided Practice: Example 3, continued Evaluate the function.
f(x) = 32(2)x
Original function
f(3) = 32(2)3
Substitute 3 for x.
f(3) = 32(8)
Simplify as needed.
f(3) = 256
3.1.4: Function Notation and Evaluating Functions

15. Guided Practice: Example 3, continued Interpret the results.
On day 3, the petition has 256 signatures. This is a point on the graph, (3, 256), of the function
f(x) = 32(2)x.
3.1.4: Function Notation and Evaluating Functions

16. Guided Practice: Example 3, continued
Number of signatures
Days ✔
3.1.4: Function Notation and Evaluating Functions

17. Guided Practice: Example 3, continued 17
3.1.4: Function Notation and Evaluating Functions