Functions

MATH 109 - PrecalculusS. Rook

Overview

• Section 1.4 in the textbook:– Relations & Functions– Functional notation– Identifying functions– Application: Difference Quotient

2

Relations & Functions

Relations vs. Functions

• Relation: any set of ordered pairs or coordinates• Function: a special type of relation where every

value of x corresponds to only one value of y– Each input produces only one output

• All functions are relations, but all relations areNOT functions:– Ex: {(0, 7), (0, -4)} is a relation, but NOT a function

4

Domain & Range

• Domain: the set of defined x-values or input values– Can sometimes be expressed as an interval

• Range: the set of defined y-values or output values– Can sometimes be expressed as an interval

• Thus far, we have seen equations in the form y = 4x– x is known as the independent variable and y is known as

the dependent variable• The independent variable (x in this case) can assume any

value• The dependent variable (y in this case) is determined by

the value chosen for x5

Domain & Range (Example)

Ex 1: Identify the i) domain ii) range for the function:

a) {(0, 4), (-2, -1), (1, 1), (7, 2)}

b)

6

Functional Notation

Functional Notation

• Until now, we have seen only equations with the letter y

• f(x) is called function notation and describes a function f in terms of x– f(x) means the same as y

y = 2x + 5 AND f(x) = 2x + 5 are the same equation/function

• To evaluate a function at a value, simply substitute the value into the function– Ex: For x = -3: f(-3) = 2(-3) + 5 = -1

8

Piecewise Functions• Piecewise Function: a set of multiple equations each defined over

specific intervals of x• To evaluate a piecewise function, we must decide which expression the

value of x corresponds to– Ex: Given f(x):

x = 8 corresponds to

1 – x4

x = 0 corresponds to

2x

x = 5 corresponds to

1 – x4

9

5,1

5 ,24 xx

xxxf

Evaluating a Function (Example)

Ex 2: Given f(x) = 3x2 – x + 2, evaluate:

a) f(1)b) f(4)c) f(-2)d) f(c – 1)

10

Evaluating a Piecewise Function (Example)

Ex 3: Given , evaluate:

a) g(0)b) g(1)c) g(3)

11

1,23

1 ,12

xx

xx

xg

Evaluating a Piecewise Function (Example)

Ex 4: Given , evaluate:

a) h(0)b) h(3)c) h(4)d) h(5)

12

4 ,12

42 ,7

2 ,32

tt

t

tt

th

Identifying Functions

Identifying Functions

• Recall that a function associates exactly ONE value of y for each value of x

• To identify a function given an equation:– Solve the equation for y and determine whether each

value of x yields one value of y

• To identify a function from a relation (a list of coordinate pairs):– Compare the x-coordinates and determine whether

each value of x yields one value of y– Look for repeating x-coordinates

14

Identifying Functions (Continued)

• Vertical Line Test: If there exists a vertical line that crosses a graph more than once, the graph is NOT a function– This means that there exists AT LEAST ONE value

of x that produces at least two different values of y

– Otherwise the graph is a function

15

Finding the Domain of a Function

• Recall that the domain of a function is the set of x values or the interval for which the function is defined

• NO domain restrictions on a polynomial– By definition a polynomial is defined over all real

numbers

• Types of functions to be aware of domain restrictions:– Square root (what is underneath must be ≥ 0)– Rational (denominator ≠ 0)

16

Identifying Functions (Example)

Ex 5: Select the function from either i) or ii) – explain why:

a) i) {(0, 0), (3,1), (-5,1), (9, 9)}ii) {(-4, 2), (-2, 3), (-4, -6), (5, -1)}

b) i) y – x2 = 4ii) y2 – x = 4

17

Identifying a Function (Example)

Ex 6: Select the function from either i) or ii) – explain why:

i) ii)

18

Finding the Domain of a Function (Example)

Ex 7: Find the domain of the function:

a) b)

c) Surface area of d) a cube: S = 6s2

19

1

2

x

xf 48 xxg

x

xh1

Application: Difference Quotient

Difference Quotient

• Given a function f, is called the difference quotient

• Allows us to study how f changes as we allow x to vary– You WILL see this again in Calculus– For this class, just be able to calculate the

difference quotient and leave it in simplest form

21

0,

h

h

xfhxf

Difference Quotient (Example)

Ex 8: Find the difference quotient – simplify the answer:

a)

b)

22

0, ,32

h

h

xfhxfxxxf

0,

22 ,12

h

h

ghgxxxg

Summary

• After studying these slides, you should be able to:– Understand the difference between a relation and a function– State the domain and range of a function given coordinate pairs

or a graph– Evaluate functions and piecewise functions– Identify the domain of a function in equation form– Calculate a difference quotient

• Additional Practice– See the list of suggested problems for 1.4

• Next lesson– Analyzing Graphs of Functions (Section 1.5)

23