Finding the Inverse of a Function from Its Graph

Q7. Find the inverse of the function given by the graph below:

Background

Topic: Inverse Functions

This question is testing your ability to find the inverse of a function using its graph. The inverse function, denoted as $f^{-1}(x)$, essentially "reverses" the original function, swapping the roles of $x$ and $y$.

Key Terms and Formulas:

• Inverse Function: If $f(x)$ maps $x$ to $y$, then $f^{-1}(x)$ maps $y$ back to $x$.

• Graphical Method: To find the inverse from a graph, reflect each point across the line $y = x$.

• Notation: If the original function passes through $(a, b)$, the inverse passes through $(b, a)$.

Step-by-Step Guidance

1. Identify the key points on the graph of $f(x)$. For example, the graph passes through $(-3, -2)$, $(-1, 0)$, and $(4, 2)$.

2. To find the inverse, swap the $x$ and $y$ values for each point. This means the inverse will pass through $(-2, -3)$, $(0, -1)$, and $(2, 4)$.

3. Plot these new points on a coordinate grid. The graph of the inverse function $f^{-1}(x)$ will be the reflection of $f(x)$ across the line $y = x$.

4. Draw the line $y = x$ for reference, and sketch the reflected points and the new line connecting them.

Try solving on your own before revealing the answer!

Final Answer:

The inverse function $f^{-1}(x)$ passes through the points $(-2, -3)$, $(0, -1)$, and $(2, 4)$. The graph is the reflection of the original function across the line $y = x$.

By swapping the coordinates of each point, you have constructed the inverse graph.