1. Functions and Their Representations
Section 2.1
Functions and Their Representations

2. Objectives Basic Concepts Representations of a Function
Definition of a Function
Identifying a Function
Tables, Graphs and Calculators (Optional)

3. FUNCTION NOTATION
The notation y = f(x) is called function notation. The input is x, the output is y, and the name of the function is f.
Name
y = f(x)
Output Input

4. The variable y is called the dependent variable and the variable x is called the independent variable. The expression f(4) = 28 is read “f of 4 equals 28” and indicates that f outputs 28 when the input is 4. A function computes exactly one output for each valid input. The letters f, g, and h, are often used to denote names of functions.

5. Representations of a Function
Verbal Representation (Words) Numerical Representation (Table of values) Symbolic Representation (Formula) Graphical Representation (Graph) Diagrammatic Representation (Diagram)

6. Example
Evaluate f(x) at the given value of x. f(x) = 5x – 3 x = −4 Solution f(−4) = 5(−4) – 3 = −20 – 3 = −23

7. Example
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution Verbal Representation Multiply a purchase of x dollars by 0.06 to obtain a sales tax of y dollars. Numerical Representation x
f(x)
$1.00
$0.06
$2.00
$0.12
$3.00
$0.18
$4.00
$0.24

8. Example (cont)
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution Symbolic Representation f(x) = 0.06x Graphical Representation

9. Example (cont)
Let a function f compute a sales tax of 6% on a purchase of x dollars. Use the given representation to evaluate f(3). Solution Diagrammatic Representation
1 ●
2 ●
3 ●
4 ●
● 0.06
● 0.12
● 0.18
● 0.24

10. Example
Let function f square the input x and then add 3 to obtain the output y.
Write a formula for f.
Make a table of values for f. Use x = −2, −1, 0, 1, 2.
Sketch a graph of f.
Solution
a. Formula If we square x and then add 3, we
obtain x Thus the formula is f(x) = x2 + 3.

11. Example (cont)
b. Make a table of values for f. Use x = −2, −1, 0, 1, 2. c. Sketch the graph. Solution x
f(x) −2 7 −1 4 3 1 2

12. Definition of a Function
A function receives an input x and produces exactly one output y, which can be expressed as an ordered pair: (x, y) Input Output
A relation is a set of ordered pairs, and a function is a special type of relation.

13. Function
A function f is a set of ordered pairs (x, y), where each x-value corresponds to exactly one y-value.
The domain of f is the set of all x-values, and the range of f is the set of all y-values.

14. Example Use the graph to find the function’s domain and range. Range

15. Example Use the graph to find the function’s domain and range. Range

16. Example
Use f(x) to find the domain of f. a. f(x) = 3x b. Solution a. Because we can multiply a real number x by 3, f(x) = 3x is defined for all real numbers. Thus the domain of f includes all real numbers. b. Because we cannot divide by 0, the input x = 4 is not valid. The domain of f includes all real numbers except 4, or x ≠ 4.

17. Slide 17

18. Example Determine whether the table represents a function.
f(x) 2 −6 3 4 −1 1
The table does not represent a function because the input x = 3 produces two outputs; 4 and −1.

19. Vertical Line Test
If every vertical line intersects a graph at no more than one point, then the graph represents a function.

20. Example Determine whether the graphs shown represent functions. a. b.
Does not pass the vertical line test.
The graph is NOT a function.
Passes the vertical line test.
The graph is a function.