Question

use the image to answer the question. the function $f(x)$ is labeled in black on the graph. which of the other functions is its inverse? $a(x)$, $b(x)$, or $c(x)$? explain how you know.

Explanation:

Step1: Recall Inverse Function Graph Rule

The graph of an inverse function \( f^{-1}(x) \) is the reflection of the graph of \( f(x) \) over the line \( y = x \).

Step2: Analyze the Graphs

Visually inspect the graphs of \( f(x) \), \( a(x) \), \( b(x) \), and \( c(x) \). The graph of \( c(x) \) appears to be the reflection of \( f(x) \) over the line \( y = x \), while \( a(x) \) and \( b(x) \) do not show this reflection property. For example, if we take a point \((x,y)\) on \( f(x) \), the point \((y,x)\) should lie on \( f^{-1}(x) \). Checking key points (like the vertex or intersection with axes) confirms that \( c(x) \) satisfies this reflection over \( y = x \).

Answer:

The inverse of \( f(x) \) is \( c(x) \) because the graph of an inverse function is the reflection of the original function's graph over the line \( y = x \), and \( c(x) \) shows this reflection relative to \( f(x) \).