Question

the function $f(x)$ is invertible. find $f^{-1}(-7)$.
$f^{-1}(-7)=\square$

Explanation:

Step1: Recall the definition of inverse function

The inverse function \( f^{-1}(y) = x \) if and only if \( f(x) = y \). So we need to find the \( x \)-value such that \( f(x)=-7 \).

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

Looking at the graph, we find the point where the \( y \)-coordinate is \( -7 \). By examining the grid, when \( y = -7 \), the corresponding \( x \)-coordinate (from the graph of \( f(x) \)) is \( 9 \)? Wait, no, wait. Wait, let's check again. Wait, the graph: let's see the coordinates. Wait, maybe I made a mistake. Wait, let's look at the grid. Each grid is 1 unit. Let's find the point where \( f(x)=-7 \). So we look for \( y = -7 \) on the graph of \( f(x) \), then find the \( x \) such that \( f(x)=-7 \). From the graph, when \( y=-7 \), what's \( x \)? Wait, looking at the graph, the curve: let's see, when \( x = 9 \)? Wait, no, maybe I misread. Wait, let's check the coordinates. Wait, the graph is a curve, let's see the points. Wait, maybe when \( x = 9 \), \( f(9)=-7 \)? Wait, no, let's count the grid. Wait, the \( x \)-axis: from 0 to 10, each grid is 1. The \( y \)-axis: from -10 to 10, each grid is 1. So when \( y = -7 \), we look at the graph of \( f(x) \), and find the \( x \) where \( f(x) = -7 \). Let's see, the graph of \( f(x) \) at \( y = -7 \), the \( x \) is 9? Wait, no, maybe 9? Wait, no, let's check again. Wait, maybe I made a mistake. Wait, the inverse function: \( f^{-1}(-7) \) is the \( x \) such that \( f(x) = -7 \). So we need to find \( x \) where \( f(x) = -7 \). From the graph, when \( y = -7 \), the \( x \)-coordinate is 9? Wait, no, let's see the graph. Wait, the curve: let's see, at \( x = 9 \), \( y = -7 \)? Wait, maybe. Wait, let's count the grid. Let's see, the \( y \)-value of -7 is 7 units below the origin. Then, looking at the graph, the point where \( y = -7 \) is at \( x = 9 \)? Wait, no, maybe 9? Wait, maybe I'm wrong. Wait, let's check again. Wait, the graph of \( f(x) \): when \( x = 9 \), \( f(9) = -7 \), so \( f^{-1}(-7) = 9 \)? Wait, no, wait, maybe I messed up. Wait, no, let's think again. The inverse function swaps \( x \) and \( y \). So if \( (a,b) \) is on \( f(x) \), then \( (b,a) \) is on \( f^{-1}(x) \). So we need to find the point on \( f(x) \) where \( y = -7 \), then the \( x \) of that point is the \( y \) of \( f^{-1}(-7) \). Wait, no: \( f^{-1}(-7) = x \) means \( f(x) = -7 \). So we need to find \( x \) such that \( f(x) = -7 \). So looking at the graph, find the \( x \) where \( y = -7 \). Let's look at the graph: the curve of \( f(x) \) at \( y = -7 \), the \( x \) is 9. So \( f(9) = -7 \), therefore \( f^{-1}(-7) = 9 \). Wait, but let's check the grid again. Wait, maybe I made a mistake. Wait, the \( x \)-axis: from 0 to 10, each grid is 1. The \( y \)-axis: from -10 to 10, each grid is 1. So when \( y = -7 \), the \( x \) is 9. So \( f(9) = -7 \), so \( f^{-1}(-7) = 9 \).

Answer:

\( \boxed{9} \)