Question

the function f(x) is invertible. find $f^{-1}(-9)$.\
$f^{-1}(-9)=\square$

Explanation:

Step1: Recall inverse function definition

The inverse function \( f^{-1}(y) \) gives the \( x \)-value such that \( f(x) = y \). So, to find \( f^{-1}(-9) \), we need to find \( x \) where \( f(x) = -9 \).

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

Looking at the graph, we identify the point where \( y = -9 \). By examining the grid, we see that when \( y = -9 \), the corresponding \( x \)-value (from the line's equation or visual inspection) is \( x = -5 \)? Wait, no, let's re-examine. Wait, the line: let's find the slope. From the graph, when \( x = 0 \), \( y = -3 \)? Wait, no, the y-intercept: looking at the graph, the line passes through (0, -3)? Wait, no, the graph: when \( x = 2 \), \( y = 0 \); when \( x = 0 \), \( y = -3 \)? Wait, no, maybe I misread. Wait, the function is a straight line? Wait, the purple line: let's check points. At \( x = 0 \), \( y = -3 \)? Wait, no, the graph shows that when \( x = 2 \), \( y = 0 \); when \( x = -4 \), what's \( y \)? Wait, maybe the function is linear. Let's find the equation of \( f(x) \). Let's take two points: (2, 0) and (0, -3). Wait, no, when \( x = 0 \), the y-coordinate is -3? Wait, no, the graph: the line crosses the y-axis at (0, -3)? Wait, no, the original graph: looking at the grid, the line goes through (0, -3)? Wait, no, the user's graph: let's see, the line passes through (2, 0) and (0, -3)? Wait, no, maybe (0, -3) is not correct. Wait, another approach: to find \( f(x) = -9 \), we need to find \( x \) such that the point (x, -9) is on \( f(x) \). Let's look at the graph: when \( y = -9 \), what's \( x \)? Let's see the slope. Let's assume the function is linear. Let's find the slope between (2, 0) and (0, -3): slope \( m = \frac{0 - (-3)}{2 - 0} = \frac{3}{2} \). So equation is \( y = \frac{3}{2}x - 3 \). Now, set \( y = -9 \): \( -9 = \frac{3}{2}x - 3 \). Solve for \( x \): \( \frac{3}{2}x = -9 + 3 = -6 \), so \( x = -6 \times \frac{2}{3} = -4 \)? Wait, no, that's not matching. Wait, maybe the y-intercept is -3? Wait, no, maybe I made a mistake. Wait, let's re-express: the inverse function \( f^{-1}(-9) \) is the \( x \) where \( f(x) = -9 \). So we need to find \( x \) such that \( f(x) = -9 \). Looking at the graph, when \( y = -9 \), what is \( x \)? Let's check the grid. The graph: each grid square is 1 unit. So, starting from the origin, moving down 9 units (y = -9), then find the x-coordinate where the line passes through y = -9. Let's see the line: from the graph, when \( x = -4 \), what's y? Wait, maybe the line has a slope of \( \frac{3}{2} \). Wait, when \( x = 2 \), \( y = 0 \); when \( x = 0 \), \( y = -3 \); when \( x = -2 \), \( y = -6 \); when \( x = -4 \), \( y = -9 \). Ah! There we go. So when \( x = -4 \), \( y = -9 \). So \( f(-4) = -9 \). Therefore, by the definition of inverse function, \( f^{-1}(-9) = -4 \).

Answer:

\( -4 \)