combinations of functions section 1.4. sum, difference, product, & quotient of functions let f...

Combinations of Combinations of FunctionsFunctions

Combinations of Combinations of FunctionsFunctions

Section 1.4Section 1.4Section 1.4Section 1.4

Sum, Difference, Product, & Quotient of Sum, Difference, Product, & Quotient of FunctionsFunctions

Sum, Difference, Product, & Quotient of Sum, Difference, Product, & Quotient of FunctionsFunctions

Let f & g be 2 functions with overlapping domains. then, for all x common to both domains, the sum, difference, product, and quotient of f & g are defined as:

1. Sum ( f+g )(x) = f (x) + g (x)

2. Difference ( f - g )(x) = f (x) - g (x)

3. Product ( f • g )(x) = f (x) • g (x)

4. Quotient ( f / g )(x) = f (x) / g (x), g(x) ≠ 0

Let f & g be 2 functions with overlapping domains. then, for all x common to both domains, the sum, difference, product, and quotient of f & g are defined as:

1. Sum ( f+g )(x) = f (x) + g (x)

2. Difference ( f - g )(x) = f (x) - g (x)

3. Product ( f • g )(x) = f (x) • g (x)

4. Quotient ( f / g )(x) = f (x) / g (x), g(x) ≠ 0

Example 1Example 1Example 1Example 1

Given f(x) = 2x + 1 & g(x) = x2 + 2x - 1

Find the sum and difference of the functions.

( f + g)(x) = f(x) + g(x) = (2x+1) + (x2 + 2x - 1) = x2 + 4x

( f - g)(x) = f(x) - g(x) = (2x+1) - (x2 + 2x - 1) = -x2 + 2

Given f(x) = 2x + 1 & g(x) = x2 + 2x - 1

Find the sum and difference of the functions.

( f + g)(x) = f(x) + g(x) = (2x+1) + (x2 + 2x - 1) = x2 + 4x

( f - g)(x) = f(x) - g(x) = (2x+1) - (x2 + 2x - 1) = -x2 + 2

Example 1 cont...Example 1 cont...Example 1 cont...Example 1 cont...

Given f(x) = 2x + 1 & g(x) = x2 + 2x - 1

Find the product of the functions.

Given f(x) = 2x + 1 & g(x) = x2 + 2x - 1

Find the product of the functions.

Example 2Example 2Example 2Example 2

Given f(x) = x + 1 & g(x) = x2 - 2x - 3

Find the quotient of the functions.

The domain of g(x) is all #’s except 3, -1. Even though it is canceled, this is still the domain.

Given f(x) = x + 1 & g(x) = x2 - 2x - 3

Find the quotient of the functions.

The domain of g(x) is all #’s except 3, -1. Even though it is canceled, this is still the domain.

Example 2Example 2Example 2Example 2

Given &

Find the quotient of the functions.

The domain of f(x) is all #’s except -1. Even though we canceled, this is still the domain.

Given &

Find the quotient of the functions.

The domain of f(x) is all #’s except -1. Even though we canceled, this is still the domain.

You TryYou TryYou TryYou Try

Given f(x) = 3x - 2 & g(x) = 3x2 + x - 2

Find the sum and difference of the functions.

( f + g)(x) = f(x) + g(x) =

(3x - 2) + (3x2 + x - 2) = 3x2 + 4x - 4

( f - g)(x) = f(x) - g(x) =

(3x - 2) - (3x2 + x - 2) = -3x2 + 2x

Given f(x) = 3x - 2 & g(x) = 3x2 + x - 2

Find the sum and difference of the functions.

( f + g)(x) = f(x) + g(x) =

(3x - 2) + (3x2 + x - 2) = 3x2 + 4x - 4

( f - g)(x) = f(x) - g(x) =

(3x - 2) - (3x2 + x - 2) = -3x2 + 2x

You TryYou TryYou TryYou Try

Given f(x) = 3x - 2 & g(x) = 3x2 + x - 2

Find the product & quotient of the functions.

( f • g)(x) = f(x) • g(x) =

(3x - 2)(3x2 + x - 2) = 9x3 -3x2 -8x + 4

(f ÷ g)(x) = f(x)/g(x) =

(3x-2)/(3x2 + x - 2) = 1/(x+1)

The domain is still x ≠ 2/3, -1

Given f(x) = 3x - 2 & g(x) = 3x2 + x - 2

Find the product & quotient of the functions.

( f • g)(x) = f(x) • g(x) =

(3x - 2)(3x2 + x - 2) = 9x3 -3x2 -8x + 4

(f ÷ g)(x) = f(x)/g(x) =

(3x-2)/(3x2 + x - 2) = 1/(x+1)

The domain is still x ≠ 2/3, -1

Composition of FunctionsComposition of FunctionsComposition of FunctionsComposition of FunctionsDefinition of The Composition of Two Functions.

The composition of the function f with the function

g is

(f∘g)(x)

The domain of (f∘g)(x) is the set of all x in the

domain of g such that g(x) is in the domain of f.

Definition of The Composition of Two Functions.

The composition of the function f with the function

g is

(f∘g)(x)

The domain of (f∘g)(x) is the set of all x in the

domain of g such that g(x) is in the domain of f.

Example 3Example 3Example 3Example 3

Given f(x) = x + 2 & g(x) = 4- x2, find the following.

a) f(g(x)) =

(4- x2) + 2 = 6 - x2

This means go to the f function and replace x with the g function.

Given f(x) = x + 2 & g(x) = 4- x2, find the following.

a) f(g(x)) =

(4- x2) + 2 = 6 - x2

This means go to the f function and replace x with the g function.

Example 3Example 3Example 3Example 3

Given f(x) = x + 2 and g(x) = 4 - x2, find the following.

a) (g ∘f)(x)

This means go to the g function and replace x with the f function.

• g(f(x)) =

4 - (x + 2)2 = 4-(x2 + 4x + 4) = -x2 - 4x

Given f(x) = x + 2 and g(x) = 4 - x2, find the following.

a) (g ∘f)(x)

This means go to the g function and replace x with the f function.

• g(f(x)) =

4 - (x + 2)2 = 4-(x2 + 4x + 4) = -x2 - 4x

Example 4 cont..Example 4 cont..Example 4 cont..Example 4 cont..

Given f(x) = x + 2 and g(x) = 4 - x2, find the following.

b) (g ∘f)(-2)

This means go to the f function & replace x with -2. Take that answer & put it into g.

• f(-2) = -2 + 2 = 0

• g(0) = 4 - 02 = 4

Given f(x) = x + 2 and g(x) = 4 - x2, find the following.

b) (g ∘f)(-2)

This means go to the f function & replace x with -2. Take that answer & put it into g.

• f(-2) = -2 + 2 = 0

• g(0) = 4 - 02 = 4

Example 4 cont..Example 4 cont..Example 4 cont..Example 4 cont..

Given f(x) = x + 2 and g(x) = 4 - x2, find the following.

b) (g ∘f)(-2)

You could also go to your answer for (g ∘f)(x) & replace x with a -2

• g( f(x)) =-x2 - 4x = -(-2)2 - 4(-2) = 4

Given f(x) = x + 2 and g(x) = 4 - x2, find the following.

b) (g ∘f)(-2)

You could also go to your answer for (g ∘f)(x) & replace x with a -2

• g( f(x)) =-x2 - 4x = -(-2)2 - 4(-2) = 4

You TryYou TryYou TryYou TryGiven f(x) = x - 2 & g(x) = x2 + 2x, find the following.

a) (f∘g)(x)

(x2 + 2x) - 2 = x2 + 2x - 2

b) (g ∘f)(x)

(x - 2)2 + 2(x - 2) = x2 - 2x

c) (f∘g)(2)

(2)2 + 2(2) - 2 = 6

d) (g∘f)(3)

(3)2 - 2(3) = 3

Given f(x) = x - 2 & g(x) = x2 + 2x, find the following.

a) (f∘g)(x)

(x2 + 2x) - 2 = x2 + 2x - 2

b) (g ∘f)(x)

(x - 2)2 + 2(x - 2) = x2 - 2x

c) (f∘g)(2)

(2)2 + 2(2) - 2 = 6

d) (g∘f)(3)

(3)2 - 2(3) = 3

Finding the Domain of the Finding the Domain of the Composition of FunctionsComposition of FunctionsFinding the Domain of the Finding the Domain of the Composition of FunctionsComposition of Functions

Given f(x) = x2 - 9 & g(x) = √(9-x2), find the composition f(g(x)). Then find the domain of f(g(x)).

We take the f function & replace x with the g function.

f(g(x)) = ((√ 9-x2))2 - 9

= 9 - x2 - 9 = -x2

Given f(x) = x2 - 9 & g(x) = √(9-x2), find the composition f(g(x)). Then find the domain of f(g(x)).

We take the f function & replace x with the g function.

f(g(x)) = ((√ 9-x2))2 - 9

= 9 - x2 - 9 = -x2

Example 5Example 5Example 5Example 5

So the composition f(g(x)) = x2, and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition.

g(x) = √(9-x2), so its domain is where 9 - x2 ≥ 0

g(x) = √(9-x2) 9 - x2 ≥ 0

[-3, 3]

So the composition f(g(x)) = x2, and the domain of this function is all real numbers. However, this is not the domain of the composition. We need to look at the domain of g(x) to know the domain of the composition.

g(x) = √(9-x2), so its domain is where 9 - x2 ≥ 0

g(x) = √(9-x2) 9 - x2 ≥ 0

[-3, 3]