Combinations of Functions

Composite Functions

Objectives

Add, subtract, multiply, and divide functions.

Find compositions of one function with another function.

Finding a Function’s Domain

Unless a function has a stated domain, its domain is the set of real numbers.

Domain Clues● The denominator cannot be zero. Numbers that cause the denominator to be zero are called

restrictions.

● An even root cannot have a negative result within the radical sign. These, too, are called restrictions.

● Unless we have these previous two conditions resulting in restrictions, then all other

equations have a domain of all real numbers.

● All real numbers are denoted by (- ∞, ∞)

Find the domain of:

We do not have a denominator or a radical sign to deal with, therefore the domain is all real numbers,

Find the domain of:

The denominator cannot be equal to zero, therefore we must solve for

Factor the equation

Solve using the zero product property

and These are the restrictions

The domain is

Find the domain of:

The number inside the radical must be zero. Therefore, we must solve for the equation

4 𝑥≥ −12

𝑥≥ −3

The domain is

The Algebra of FunctionsThe Sum of Functions

Let f and g be two functions.

The sum of (f + g) (x) = f(x) + g(x)

Simply add the two equations and simplify.

The domain of f(x) + g(x) is the set of all real numbers common to the domains of f and g.

Let and

Find Determine the domain.

(2 𝑥−1 )+(𝑥2+𝑥− 2)

Simply add the two expressions and simplify.

What is the domain? All real numbers,

The Algebra of FunctionsThe Difference of Functions

Let f and g be two functions.

The difference of (f - g) (x) = f(x) - g(x)

Simply subtract the two equations and simplify

The domain of f(x) - g(x) is the set of all real numbers common to the domains of f and g.

Let and

Find

Determine the domain.

Simply subtract the expressions and simplify.

(2 𝑥−1 ) −(𝑥2+𝑥− 2)

2 𝑥−1−𝑥2−𝑥+2( 𝑓 −𝑔 ) (𝑥 )=−𝑥2+𝑥+1

The domain =

The Algebra of FunctionsThe Product of Functions

Let f and g be two functions.

The product of (fg) (x) = f(x) • g(x)

Simply multiply the two equations and simplify

The domain of f(x) • g(x) is the set of all real numbers common to the domains of f and g.

Let and

Find

Determine the domain.

Simply multiply the two equations.

(2 𝑥−1)(𝑥2+𝑥−2)

2 𝑥3+2𝑥2− 4 𝑥− 𝑥2 −𝑥+2

( 𝑓𝑔 ) (𝑥 )=2𝑥3+𝑥2− 5𝑥+2

The domain is

The Algebra of FunctionsThe Quotient of Functions

Let f and g be two functions.

The quotient of (f/g) (x) = f(x)/g(x)

Simply divide the two equations and simplify

The domain of f(x) / g(x) is the set of all real numbers common to the domains of f and g, provided g(x) is not equal to zero.

Let and

Find Determine the domain.

Can the function be simplified? No

Therefore, the above function is the answer.

The domain is all real numbers except for the restrictions which would result in the denominator equal to zero. The restrictions are .

Forming Composite Functions(f ◦ g) (x) = f(g(x))

Substitute the expression for g(x) into every x for the equation of f(x).

Given: f(x) = 3x – 4 and g(x) = x² - 2x + 6

Find (f ◦ g) (x).

(f ◦ g) (x) = f(g(x)), so substitute x² - 2x + 6 for x in the expression 3x – 4.

To be continued

Forming Composite FunctionsSubstitute x² - 2x + 6 for x in the expression 3x – 4.

3 (𝑥2− 2𝑥+6 ) − 4

3 𝑥2− 6𝑥+18 − 4

(f ◦ g) (x) =

Forming Composite Functions(g ◦ f) (x) = g(f(x))

Substitute the expression for f(x) into every x in the equation of g(x).

Given: f(x) = 3x – 4 and g(x) = x² - 2x + 6

Find (g ◦ f) (x).

(g ◦ f) (x) = g(f(x)), so substitute 3x – 4 for x in the expression x² - 2x + 6.

To be continued

Forming Composite FunctionsSubstitute 3x – 4 for x in the expression x² - 2x + 6.

(3 𝑥− 4 )2− 2 (3𝑥−4 )+6

9 𝑥2− 24 𝑥+16 − 6 𝑥+8+6

( g  ◦ f ) ( x ) = 9𝑥2− 30 𝑥+30

Find (g ◦ f)(1)

(g ◦ f) (x) = 9x² - 30x + 30

Substitute 1 for every x

9(1)² - 30(1) + 30

9 – 30 + 30 = 9

(g ◦ f)(1) = 9