Question

whether the function has an inverse function.
$f(x) = \sqrt{x - 4}$, $x \geq 4$
yes, $f$ does have an inverse.
no, $f$ does not have an inverse.
if it does, find the inverse function. (if an answer does not exist, enter dne.)
$f^{-1}(x) = \square$, $x \geq 0$

Explanation:

Step1: Let \( y = f(x) = \sqrt{x - 4} \) (where \( x \geq 4 \))

We start by setting \( y \) equal to the function \( f(x) \). This is the first step in finding the inverse function, as we need to express \( x \) in terms of \( y \).

Step2: Solve for \( x \) in terms of \( y \)

Square both sides of the equation \( y = \sqrt{x - 4} \) to get rid of the square root. We have \( y^{2}=x - 4 \). Then, add 4 to both sides to solve for \( x \). So, \( x=y^{2}+4 \). Since the original function has \( x\geq4 \), the range of \( f(x) \) (which will be the domain of the inverse function) is \( y\geq0 \) (because the square root of a non - negative number is non - negative).

Step3: Swap \( x \) and \( y \) to get the inverse function

To find the inverse function, we swap \( x \) and \( y \). So the inverse function \( f^{-1}(x) \) is obtained by replacing \( y \) with \( x \) and \( x \) with \( y \) in the equation \( x = y^{2}+4 \). We get \( y=x^{2}+4 \), with the domain \( x\geq0 \) (because the range of the original function \( f(x) \) is \( y\geq0 \), which becomes the domain of the inverse function). So \( f^{-1}(x)=x^{2}+4 \), \( x\geq0 \)

Answer:

\( f^{-1}(x)=x^{2}+4 \), \( x\geq0 \)