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 \( x \) such that \( f(x) = y \). So, we need to find \( x \) where \( f(x) = -2 \).

Step2: Analyze the graph of \( f(x) \)

Looking at the graph of \( f(x) \), we find the point where \( y = -2 \). From the grid, when \( y = -2 \), the corresponding \( x \)-value (by checking the coordinates on the line) is \( x = 1 \)? Wait, no, let's recheck. Wait, the line passes through (0,0) and has a slope. Wait, let's find the equation of \( f(x) \). The slope \( m \) between (0,0) and, say, (1, -2)? Wait, no, when \( x = 1 \), \( y = -2 \)? Wait, no, let's see: when \( y = -2 \), what's \( x \)? Let's look at the graph. The line goes through (0,0) and, for example, when \( x = 1 \), \( y = -2 \)? Wait, no, let's check the coordinates. Wait, the graph: when \( y = -2 \), the \( x \)-coordinate is 1? Wait, no, maybe I made a mistake. Wait, the function is a straight line. Let's find two points. When \( x = 0 \), \( y = 0 \). When \( x = 1 \), \( y = -2 \)? Wait, no, when \( x = 1 \), looking at the grid, the \( y \)-value is -2? Wait, the vertical axis is \( y \), horizontal is \( x \). So, to find \( f^{-1}(-2) \), we need \( x \) such that \( f(x) = -2 \). So we look for the \( x \) where the graph of \( f(x) \) has \( y = -2 \). From the graph, when \( y = -2 \), the \( x \) is 1? Wait, no, wait: let's see the line. Let's take two points: (0,0) and (2, -4)? Wait, no, when \( x = 1 \), \( y = -2 \)? Wait, the grid lines: each square is 1 unit. So, the line passes through (0,0) and, for example, when \( x = 1 \), \( y = -2 \)? Wait, no, when \( x = 1 \), moving up/down: the \( y \)-coordinate at \( x = 1 \) is -2? Yes, because the line goes from (0,0) downwards with slope -2? Wait, slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Let's take (0,0) and (1, -2): slope is \( \frac{-2 - 0}{1 - 0} = -2 \). So the equation is \( f(x) = -2x \). Then, to find \( f(x) = -2 \), solve \( -2x = -2 \), so \( x = 1 \). Wait, but let's check the graph again. When \( x = 1 \), \( y = -2 \), so \( f(1) = -2 \). Therefore, by the definition of inverse function, \( f^{-1}(-2) = 1 \).

Answer:

\( 1 \)