Question

the function f(x) is invertible. find $f^{-1}(-6)$. the graph of f(x) is given on a coordinate plane with x-axis from -10 to 10 and y-axis from -10 to 10. the function f(x) is a curve starting from the left, curving upwards, and approaching a horizontal asymptote at y = -2 as x increases. we need to find $f^{-1}(-6)$.

Explanation:

Step1: Recall inverse function definition

To find \( f^{-1}(-6) \), we need to find the \( x \)-value such that \( f(x) = -6 \) (since if \( f(a) = b \), then \( f^{-1}(b) = a \)).

Step2: Locate \( y = -6 \) on the graph

Look at the graph of \( f(x) \) and find the point where the \( y \)-coordinate is \( -6 \). Then, identify the corresponding \( x \)-coordinate of that point. From the graph, when \( y = -6 \), the \( x \)-coordinate is \( -2 \). So, if \( f(-2) = -6 \), then by the definition of the inverse function, \( f^{-1}(-6) = -2 \).

Answer:

\( -2 \)