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Manipulating Radical Functions

Manipulating Radical Functions 1.3 - Solution

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We want to determine whether the given pair of functions are inverse functions. f(x)=(x+6)2andg(x)=x6\begin{gathered} f(x)=(x+6)^2 \quad \text{and} \quad g(x)=\sqrt{x}-6 \end{gathered} To do so, we need to verify that the compositions of f(x)f(x) and g(x)g(x) are the identity function. Since the domain of g(x)g(x) is all real numbers greater than or equal to zero, we will only consider nonnegative values for the x-x\text{-}variable.

Calculating f(g(x))f(g(x))

To find the expression for f(g(x))f(g(x)) we will start by substituting x6\sqrt{x}-6 for g(x).g(x). f(g(x))f(x6) f(g(x)) \quad \Rightarrow \quad f \left( {\color{#0000FF}{\sqrt{x}-6}} \right) Now we apply the definition of f(x).f(x). f(x)=(x+6)2f(x6)=(x6+6)2\begin{gathered} f(x)=(x+6)^2 \\ \Downarrow \\ f\left({\color{#0000FF}{\sqrt{x}-6}} \right) = \left( {\color{#0000FF}{\sqrt{x}-6}}+6 \right)^2 \end{gathered} Finally, let's simplify and see if the function is the identity function.
f(g(x))=(x6+6)2f(g(x)) = \left( \sqrt{x}-6+6 \right) ^2
f(g(x))=(x)2f(g(x)) = \left( \sqrt{x} \right) ^2
f(g(x))=x f(g(x)) = x\ {\color{#009600}{\huge{\checkmark}}}
We found that f(g(x))f(g(x)) is the identity function.

Calculating g(f(x))g(f(x))

We will now investigate the expression g(f(x)).g(f(x)). To find this we will start by substituting (x+6)2(x+6)^2 for f(x).f(x). g(f(x))g((x+6)2) g(f(x)) \quad \Rightarrow \quad g \left( {\color{#0000FF}{(x+6)^2}} \right) Now we apply the definition of g(x).g(x). g(x)=x6g((x+6)2)=(x+6)26\begin{gathered} g(x)=\sqrt{x}-6 \\ \Downarrow \\ g\left( {\color{#0000FF}{(x+6)^2}} \right) = \sqrt{{\color{#0000FF}{(x+6)^2}}}-6 \end{gathered} Finally, let's simplify and see if the function is the identity function.
g(f(x))=(x+6)26g(f(x)) = \sqrt{(x+6)^2}-6
g(f(x))=x+66g(f(x)) = |x+6|-6
As we said at the beginning of the exercise, we are only considering nonnegative values for the x-x\text{-}variable. Thus, x+6|x+6| == x+6.x+6.
g(f(x))=x+66g(f(x)) = |x+6|-6
g(f(x))=x+66g(f(x)) = x+6-6
g(f(x))=x g(f(x)) = x\ {\color{#009600}{\huge{\checkmark}}}
We found that g(f(x))g(f(x)) is also the identity function. Thus, f(x)f(x) and g(x)g(x) are inverse functions.