Question

the function f(x) is invertible. find $f^{-1}(-6)$.
$f^{-1}(-6) = \square$

Explanation:

Step1: Recall inverse function property

For an invertible function \( f \), if \( f(a) = b \), then \( f^{-1}(b) = a \). So we need to find \( x \) such that \( f(x) = -6 \).

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

Looking at the graph, find the point where \( y = -6 \) (since \( f(x) = y \)). The graph of \( f(x) \) intersects the \( y \)-axis (where \( x = 0 \)) at \( (0, -6) \)? Wait, no, wait. Wait, the graph: let's check the coordinates. Wait, the line: when \( x = 0 \), what's \( y \)? Wait, the graph shows that at \( x = 0 \), \( y = -6 \)? Wait, no, wait the grid. Wait, the y-axis: the grid lines are at integers. Wait, the line passes through (0, -6)? Wait, no, wait the arrow. Wait, the function \( f(x) \) is a line. Let's see, when \( x = 0 \), the \( y \)-value is -6? Wait, no, looking at the graph, the line goes through (0, -6)? Wait, no, wait the coordinates. Wait, the graph: the red line. Let's check the points. Wait, when \( x = 0 \), \( y = -6 \)? Wait, no, maybe I misread. Wait, the problem is to find \( f^{-1}(-6) \), which is the \( x \) such that \( f(x) = -6 \). So we need to find the \( x \)-value where \( f(x) = -6 \). Looking at the graph, the line \( f(x) \) intersects the point (0, -6)? Wait, no, wait the graph: when \( x = 0 \), \( y = -6 \)? Wait, the y-axis: the vertical axis. So at \( x = 0 \), \( y = -6 \). So \( f(0) = -6 \). Therefore, by the definition of inverse function, \( f^{-1}(-6) = 0 \).

Answer:

\( \boxed{0} \)