Question3. The graph of $y=f^{-1}(x)$ is shown. Evaluate $f^{-1}(2)$

Studdy Solution

STEP 1

1. The function $f^{-1}(x)$ represents the inverse of some function $f(x)$ .
2. The graph provided is that of $y = f^{-1}(x)$ .
3. The task is to find the value of $f^{-1}(2)$ by examining the graph.

STEP 2

1. Understand the relationship between $f(x)$ and $f^{-1}(x)$ .
2. Locate the point $x = 2$ on the graph of $f^{-1}(x)$ .
3. Determine the corresponding $y$ -value for $x = 2$ .
4. Conclude $f^{-1}(2)$ based on the graph.

STEP 3

Recall that the inverse function $f^{-1}(x)$ satisfies the condition $y = f^{-1}(x)$ if and only if $x = f(y)$ . This means that for $y = f^{-1}(x)$ , the roles of $x$ and $y$ are reversed compared to $y = f(x)$ .

STEP 4

Locate the point where $x = 2$ on the graph of $y = f^{-1}(x)$ . This involves finding the vertical line $x = 2$ on the graph.

STEP 5

Identify where the vertical line $x = 2$ intersects the graph of $y = f^{-1}(x)$ .

STEP 6

Read off the corresponding $y$ -value at the point of intersection from the graph. This $y$ -value is the value of $f^{-1}(2)$ .

STEP 7

From the image description, we observe that the graph intersects the line $x = 2$ at $y = 1$ .

STEP 8

Conclude that $f^{-1}(2) = 1$ based on the graph.
Solution: The value of $f^{-1}(2)$ is 1.

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