Exercise 8 Page 414

We want to determine whether the given pair of functions are inverse functions. f(x)=2x+16 and g(x)= 12x-8 To do so, we need to verify that the compositions of f(x) and g(x) are the identity function.

Calculating [f ∘ g](x)

Recall that [f ∘ g](x) = f(g(x)). To find the expression, we will start by substituting 12x-8 for g(x).

[f ∘ g](x) = f ( g(x) )

Substitute

g(x)= 1/2x-8

[f ∘ g](x) = f ( 1/2x-8 )

Now we apply the definition of f(x). f(x)=2x+16 ⇓ f( 12x-8 ) = 2( 12x-8 ) + 16 Finally, let's simplify and see if the function is the identity function.

[f ∘ g](x) = 2( 1/2x-8) + 16

Distr

Distribute 2

[f ∘ g](x) = x - 16 + 16

AddTerms

Add terms

[f ∘ g](x) = x ✓

We found that [f ∘ g](x) is the identity function.

Calculating [g ∘ f](x)

Similarly, recall that [g ∘ f](x) = g( f(x) ). To find the expression, this time we will start by substituting 2x+16 for f(x).

[g ∘ f](x) = g ( f(x) )

Substitute

g(x)= 2x+16

[g ∘ f](x) = g ( 2x+16 )

Now we apply the definition of g(x). g(x)= 12x-8 ⇓ g( 2x+16 ) = 12( 2x+16)-8 Finally, let's simplify and see if the function is the identity function.

[g ∘ f](x) = 1/2(2x+16)-8

Distr

Distribute 12

[g ∘ f](x) = x+8-8

SubTerms

Subtract terms

[g ∘ f](x) = x ✓

We found that [g ∘ f](x) is also the identity function. As such, f(x) and g(x) are inverse functions.