Question

consider this function.
$f(x)=|x - 4|+6$
if the domain is restricted to the portion of the graph with a positive slope, how are the domain and range of the function and its inverse related?
since the domain of the original function is limited to $x\geq6$, the range of the inverse function is $y\leq6$.
since the domain of the original function is limited to $x\geq4$, the range of the inverse function is $y\leq1$.
since the range of the original function is limited to $y\geq6$, the domain of the inverse function is $x\geq6$.
since the range of the original function is limited to $y\geq4$, the domain of the inverse function is $x\geq1$.

Explanation:

Step1: Analyze the original function

The function \(f(x)=|x - 4|+6\). The graph of \(y = |x - 4|+6\) is a V - shaped graph with vertex at \((4,6)\). The function \(y=|x - 4|+6=

$$\begin{cases}x - 4+6=x + 2, &x\geq4\\-(x - 4)+6=-x+10, &x<4\end{cases}$$

\). The part of the graph with positive slope is \(y=x + 2\) for \(x\geq4\).

Step2: Recall the property of inverse functions

For a one - to - one function \(y = f(x)\) and its inverse \(x = f^{-1}(y)\), the domain of \(f\) is the range of \(f^{-1}\) and the range of \(f\) is the domain of \(f^{-1}\). For \(y=f(x)=x + 2\) where \(x\geq4\), the range of \(f(x)\) is \(y\geq6\) (when \(x = 4,y=4 + 2=6\) and as \(x\) increases, \(y\) increases).

Answer:

Since the range of the original function is limited to \(y\geq6\), the domain of the inverse function is \(x\geq6\)