Exercise 1 Page 443

We want to determine whether the given pair of functions are inverse functions. f(x)=3x+8 and g(x)= x-83 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 x-83 for g(x).

[f ∘ g](x) = f ( g(x) )

Substitute

g(x)= x-8/3

[f ∘ g](x) = f ( x-8/3 )

Now we apply the definition of f(x). f(x)=3x+8 ⇓ f( x-8/3) = 3( x-8/3)+8 Finally, let's simplify and see if the function is the identity function.

[f ∘ g](x) = 3(x-8/3)+8

▼

Simplify

MoveLeftFacToNum

a*b/c= a* b/c

[f ∘ g](x) = 3(x-8)/3+8

CalcQuot

Calculate quotient

[f ∘ g](x) = x-8+8

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 3x+8 for f(x).

[g ∘ f](x) = g ( f(x) )

Substitute

g(x)= 3x+8

[g ∘ f](x) = g ( 3x+8 )

Now we apply the definition of g(x). g(x)=x-8/3 ⇓ g( 3x+8 ) = ( 3x+8)-8/3 Finally, let's simplify and see if the function is the identity function.

[g ∘ f](x) = (3x+8)-8/3

▼

Simplify

RemovePar

Remove parentheses

[g ∘ f](x) = 3x+8-8/3

AddSubTerms

Add and subtract terms

[g ∘ f](x)=3x/3

CalcQuot

Calculate quotient

[g ∘ f](x) = x ✓

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