Question

determine whether the function has an inverse function.
$f(x) = |x - 7|$, $x \leq 7$
$\circ$ yes, $f$ does have an inverse.
$\circ$ 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: Analyze the function's domain and simplify

The function is \( f(x) = |x - 7| \) with \( x \leq 7 \). For \( x \leq 7 \), \( x - 7 \leq 0 \), so \( |x - 7|=-(x - 7)=7 - x \). So \( f(x)=7 - x \) when \( x \leq 7 \). This is a linear function with a slope of -1, which is one - to - one (since it's a straight line with non - zero slope, and for a linear function \( y = mx + b\), \( m
eq0\) implies it's one - to - one). A one - to - one function has an inverse.

Step2: Find the inverse function

Let \( y = f(x)=7 - x \), where \( x \leq 7 \) and \( y\geq0 \) (since when \( x = 7\), \( y = 0\); as \( x\) decreases, \( y = 7 - x\) increases).
To find the inverse, we solve for \( x\) in terms of \( y\):
Start with \( y=7 - x\).
Add \( x\) to both sides: \( y + x=7\).
Subtract \( y\) from both sides: \( x = 7 - y\).
Then, we interchange \( x\) and \( y\) to get the inverse function. So \( f^{-1}(x)=7 - x\), with the domain of the inverse function being \( x\geq0\) (since the range of \( f(x)\) for \( x\leq7\) is \( y\geq0\), and the domain of the inverse function is the range of the original function).

Answer:

Yes, \( f \) does have an inverse. \( f^{-1}(x)=7 - x\), \( x\geq0\)