Determine whether the function has an inverse function, If it does, find its inverse function. $f ( x ) = | x - 2 | , x \leq 2$
A)No, $f$ does not have an inverse function.
B)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = 2 - x , x \geq 0$
C)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - 2 - x , x \geq 0$
D)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = 2 + x , x \geq 0$
E)Yes, $f$ has an inverse function, $f ^ { - 1 } ( x ) = - 2 + x , x \geq 0$

Question

Quizplus · Accepted Answer

Given $f(x) = |x - 2|$ for $x \leq 2$, the function simplifies to $f(x) = 2 - x$ because for $x \leq 2$, $x - 2 \leq 0$ and the absolute value of a negative number is its positive counterpart. The inverse function swaps $x$ and $y$, leading to $x = 2 - y$, which simplifies to $y = 2 - x$. Since $f(x)$ is decreasing and passes the horizontal line test, it has an inverse, which is $f^{-1}(x) = 2 - x$, valid for $x \geq 0$ because $f(x)$ outputs non-negative values.

Determine Whether the Function Has an Inverse Function, If It f(x)=∣x−2∣,x≤2f ( x ) = | x - 2 | , x \leq 2f(x)=∣x−2∣,x≤2

Determine Whether the Function Has an Inverse Function, If It $f ( x ) = | x - 2 | , x \leq 2$