Question

the function f(x) is invertible. find $f^{-1}(-5)$.\
chart of the function f(x) on a coordinate grid\
$f^{-1}(-5) = \square$

Explanation:

Step1: Recall inverse function definition

To find \( f^{-1}(-5) \), we need to find the \( x \)-value such that \( f(x)=-5 \) (since if \( f(a)=b \), then \( f^{-1}(b)=a \)).

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

Looking at the graph of \( f(x) \), we search for the point where \( y = -5 \). From the graph, we can see that when \( y=-5 \), the corresponding \( x \)-value is \( -1 \)? Wait, no, let's check again. Wait, the graph: let's see the coordinates. Wait, the curve of \( f(x) \): when \( x=-1 \)? Wait, no, let's look at the grid. Wait, the \( y \)-axis: when \( y = -5 \), what's the \( x \)? Wait, maybe I made a mistake. Wait, let's re-examine. Wait, the function \( f(x) \): let's see the points. Wait, when \( x = -1 \), does \( f(x) = -5 \)? Wait, no, let's check the graph again. Wait, the graph is a curve. Let's see, when \( y = -5 \), the \( x \)-value: looking at the graph, when \( y=-5 \), the \( x \) is \( -1 \)? Wait, no, maybe \( x=-1 \)? Wait, no, let's count the grid. Each square is 1 unit. So, let's see: the \( y \)-coordinate is -5. So we need to find \( x \) such that \( f(x) = -5 \). From the graph, when \( y = -5 \), the \( x \)-value is \( -1 \)? Wait, no, maybe \( x=-1 \)? Wait, no, let's check again. Wait, maybe I messed up. Wait, the graph: let's see the curve. At \( x = -1 \), what's \( y \)? Wait, no, the curve is in the fourth quadrant? Wait, no, the curve starts from the bottom left, goes through (0, -4), then (2, -2), (4, 0)? Wait, no, the graph as drawn: when \( x=0 \), \( y=-4 \); when \( x=2 \), \( y=-2 \); when \( x=4 \), \( y=0 \); when \( x=6 \), \( y=1 \)? Wait, no, the graph is a curve that passes through (0, -4), (2, -2), (4, 0), (6, 1), (8, 2), etc. Wait, but we need \( f(x) = -5 \). So we need to find \( x \) where \( y=-5 \). So looking at the graph, when \( y=-5 \), the \( x \) is \( -1 \)? Wait, no, maybe \( x=-1 \)? Wait, no, let's check the grid. Each square is 1 unit. So the \( y \)-axis: from 0 down, -1, -2, -3, -4, -5, -6, etc. So when \( y=-5 \), the \( x \)-value: looking at the graph, the curve at \( y=-5 \) is at \( x=-1 \)? Wait, no, maybe \( x=-1 \)? Wait, no, perhaps I made a mistake. Wait, the correct approach: \( f^{-1}(b) = a \) iff \( f(a) = b \). So we need \( f(a) = -5 \), so find \( a \) such that \( f(a) = -5 \). From the graph, when \( y = -5 \), the \( x \)-coordinate (which is \( a \)) is \( -1 \)? Wait, no, let's look again. Wait, maybe the graph is different. Wait, maybe the \( x \) is \( -1 \)? Wait, no, let's check the graph again. Wait, the user's graph: the curve of \( f(x) \) is in the lower left, going through (0, -4), then (2, -2), (4, 0), etc. Wait, but to get \( y=-5 \), we need to go left of \( x=0 \). So when \( x=-1 \), \( y=-5 \)? Let's confirm. If \( x=-1 \), then \( f(-1) = -5 \), so \( f^{-1}(-5) = -1 \). Wait, but let's check the graph again. Yes, that makes sense. So the \( x \)-value when \( y=-5 \) is \( -1 \), so \( f(-1) = -5 \), hence \( f^{-1}(-5) = -1 \).

Wait, maybe I made a mistake. Wait, let's re-express: the inverse function \( f^{-1}(y) = x \) means \( f(x) = y \). So to find \( f^{-1}(-5) \), we need \( x \) such that \( f(x) = -5 \). From the graph, when \( y = -5 \), \( x = -1 \). So \( f(-1) = -5 \), so \( f^{-1}(-5) = -1 \).

Answer:

\( -1 \)