Exercise 12 Page 225

We want to determine whether the given pair of functions are inverse functions. f(x)=x-6 and g(x)=x+6 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+6 for g(x).

[f ∘ g](x) = f ( g(x) )

Substitute

g(x)= x+6

[f ∘ g](x) = f ( x+6 )

Now we apply the definition of f(x). f(x)=x-6 ⇓ f( x+6) = x+6-6 Finally, let's simplify and see if the function is the identity function.

[f ∘ g](x) = x+6-6

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[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 x-6 for f(x).

[g ∘ f](x) = g ( f(x) )

Substitute

f(x)= x-6

[g ∘ f](x) = g ( x-6 )

Now we apply the definition of g(x). g(x)=x+6 ⇓ g( x-6 ) = x-6+6 Finally, let's simplify and see if the function is the identity function.

[g ∘ f](x) = x-6+6

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[g ∘ f](x) = x ✓

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