Question

Given the function F(x)= square root(x-12) complete parts A, B, and C

Answer 1

Given the function f(x) defined as:

$f(x)=\sqrt[]{x-12}$

(a)

To find the inverse of f(x), we express the function as:

$y=\sqrt[]{x-2}$

Now, we take the square on both sides:

$\begin{gathered} y^2=x-12 \\ \Rightarrow x=y^2+12 \end{gathered}$

We change the notation:

$\begin{gathered} x\rightarrow f^{-1}(x) \\ y\rightarrow x \end{gathered}$

Then, the inverse function is:

$f^{-1}(x)=x^2+12$

For x ≥ 0

(c)

The domain of f(x) are those values such that:

$\begin{gathered} x-12\ge0\Rightarrow x\ge12 \\ \text{Dom}_f=\lbrack12,\infty) \end{gathered}$

And the range is the set of all positive numbers (including 0):

$\text{Ran}_f=\lbrack0,\infty)$

For the inverse, the domain of f(x) is its range, and the range of f(x) is its domain:

$\begin{gathered} \text{Dom}_{f^{-1}}=\lbrack0,\infty) \\ \text{Ran}_{f^{-1}}=\lbrack12,\infty) \end{gathered}$