Composition of Functions HW Algebra 2

Question

Given 𝒇(𝒙)=𝒙+𝟒 and 𝒈(𝒙)=𝒙𝟐+𝟐𝒙+𝟏, find each function or value. 

 

1) [f∘g](x)  

2) [g∘f](x)

3) [f∘g](6)          

4) [g∘f](−3)

Use the table to find each value 

 

5) [g∘f](0)

6) [h∘g](−2)

  

x f(x). g(x). h(x)

-2. -2. 1. 0

-1. -4 1 0

0. -2. -2 -2

1. 2. -1 -2

2. 1 0 5

Answer 1

Given f(x) = x + 4 and g(x) = x² + 2x + 1

1) [f ∘ g](x) = f [ g(x) ]

= f(x² + 2x + 1) *first we plug in our g(x)

= (x² + 2x + 1) + 4 *then we plug in our g(x) equation into our f(x) equation

= x² + 2x + 5

2) [g ∘ f](x) = g [ f(x) ]

= g(x + 4) *first we plug in our f(x)

= (x + 4)² + 2(x + 4) + 1 *then we plug in our f(x) equation into out g(x) equation.

Foil and distribute

= x² + 4x + 4x + 16 + 2x + 8 + 1 *combine your like terms

= x² + 10x + 25

3) [f ∘ g](6) = f [ g(6) ]

= f [6² + 2(6) + 1]

= f[36 + 12 + 1]

= f[49]

=49 + 4

= 53

4) [g ∘ f](−3) = g[ f(-3) ]

= g[(-3)² + 2(-3) + 1]

= g[9 - 6 + 1]

= g[4]

= 4 + 4

= 8

5) [g ∘ f ](0) = g [ f(0) ]

Look at the table. If we plug in 0 for x, we look at x = 0 and see that under f(x) it equals -2.

f(0) = -2

We plug this in so then it becomes g(-2).

Then we look at the table again and see that when x = -2, we follow across under g(x) and see that it equals 1. So g(-2) = 1

Hence, [g ∘ f ](0) = g [ f(0) ] = 1

6) [h ∘ g ](-2) = h [ g(-2) ]

Look at the table. If we plug in -2 for x, we look at x = -2 and see that under g(x) it equals 1.

g(-2) = 1

We plug this in so then it becomes h(1).

Then we look at the table again and see that when x = 1,

we follow across under h(x) and see that it equals -2.

So g(-2) = 1

Hence, [h ∘ g ](-2) = h [ g(-2) ] = 1