Question

the function f(x) is invertible. find $f^{-1}(-2)$.
$f^{-1}(-2) = \square$

Explanation:

Step1: Recall inverse function definition

The inverse function \( f^{-1}(y) \) gives the \( x \)-value such that \( f(x) = y \). So, to find \( f^{-1}(-2) \), we need to find \( x \) where \( f(x) = -2 \).

Step2: Analyze the graph

Look at the graph of \( f(x) \). Find the point on the graph where the \( y \)-coordinate is \( -2 \). From the graph, when \( y = -2 \), the corresponding \( x \)-value is \( -2 \)? Wait, no, let's check again. Wait, the graph: let's see the curve. Wait, when \( y = -2 \), what's \( x \)? Wait, looking at the grid, when \( y = -2 \), the \( x \)-value is \( -2 \)? Wait, no, wait the graph: the curve is at \( x = -2 \) when \( y = -2 \)? Wait, no, let's look at the coordinates. Wait, the graph: when \( y = -2 \), the \( x \) is \( -2 \)? Wait, no, maybe I misread. Wait, the graph: let's see, the curve goes through \( x = -2 \), \( y = -2 \)? Wait, no, let's check the grid. The \( x \)-axis and \( y \)-axis: each grid is 1 unit. So when \( y = -2 \), the \( x \)-coordinate is \( -2 \)? Wait, no, wait the graph: the blue curve, when \( y = -2 \), what's \( x \)? Wait, maybe I made a mistake. Wait, the inverse function: \( f^{-1}(-2) \) is the \( x \) such that \( f(x) = -2 \). So find \( x \) where \( f(x) = -2 \). From the graph, looking at the curve, when \( y = -2 \), the \( x \) is \( -2 \)? Wait, no, let's check again. Wait, the graph: the curve is at \( x = -2 \), \( y = -2 \)? Wait, no, maybe \( x = -2 \) when \( y = -2 \)? Wait, no, let's see the coordinates. Let's look at the grid: the \( x \)-axis has -10, -8, -6, -4, -2, 0, 2, etc. The \( y \)-axis has -10, -8, -6, -4, -2, 0, 2, etc. The curve: when \( y = -2 \), the \( x \) is \( -2 \)? Wait, no, maybe \( x = -2 \) is when \( y = -2 \). Wait, let's confirm: \( f(-2) = -2 \), so \( f^{-1}(-2) = -2 \)? Wait, no, that can't be. Wait, maybe I misread. Wait, the graph: let's see, the curve is at \( x = -2 \), \( y = -2 \)? Wait, no, maybe \( x = -2 \) is the \( x \) when \( y = -2 \). So \( f(-2) = -2 \), so \( f^{-1}(-2) = -2 \)? Wait, no, that seems off. Wait, maybe I made a mistake. Wait, let's think again. The inverse function swaps \( x \) and \( y \). So the point \( (a, b) \) on \( f(x) \) corresponds to \( (b, a) \) on \( f^{-1}(x) \). So we need to find the \( x \) such that \( f(x) = -2 \), which is the \( x \) where \( (x, -2) \) is on \( f(x) \). Looking at the graph, the point \( (-2, -2) \) is on \( f(x) \)? Wait, no, let's check the graph again. Wait, the curve: when \( x = -2 \), \( y = -2 \)? Wait, maybe. So \( f(-2) = -2 \), so \( f^{-1}(-2) = -2 \). Wait, but let's check again. Wait, the graph: the blue curve, at \( x = -2 \), \( y = -2 \). So yes, \( f(-2) = -2 \), so \( f^{-1}(-2) = -2 \).

Wait, no, maybe I messed up. Wait, the graph: let's see, the curve is going from the bottom left, curving up to (0, -1) or something? Wait, no, the graph is a curve that at \( x = -2 \), \( y = -2 \)? Wait, maybe the correct \( x \) is \( -2 \). So \( f^{-1}(-2) = -2 \).

Answer:

\(-2\)