Question13. A function $f(x)$ is one-to-one. If the graph of $f^{-1}(x)$ lies in the fourth quadrant. In which quadrant does the graph of $f(x)$ lie? A. First Quadrant B. Second Quadrant b) Given the graph of the function $h(x)$ . graph $h^{-1}(x)$ C. Third Quadrant D. Fourth Quadrant

Studdy Solution

STEP 1

1. The function $f(x)$ is one-to-one, meaning it has an inverse function $f^{-1}(x)$ .
2. The graph of $f^{-1}(x)$ lies in the fourth quadrant.
3. We need to determine in which quadrant the graph of $f(x)$ lies.

STEP 2

1. Understand the relationship between a function and its inverse.
2. Determine the quadrant of $f(x)$ based on the quadrant of $f^{-1}(x)$ .

STEP 3

Understand the relationship between a function and its inverse.
- A function and its inverse are reflections of each other across the line $y = x$ . - If a point $(a, b)$ lies on the graph of $f(x)$ , then the point $(b, a)$ lies on the graph of $f^{-1}(x)$ .

STEP 4

Determine the quadrant of $f(x)$ based on the quadrant of $f^{-1}(x)$ .
- The fourth quadrant is defined by positive $x$ -values and negative $y$ -values. - If $f^{-1}(x)$ lies in the fourth quadrant, then it contains points of the form $(a, -b)$ , where $a > 0$ and $b > 0$ . - Reflecting these points across the line $y = x$ , the points on $f(x)$ will be $(-b, a)$ . - Therefore, $f(x)$ will lie in the second quadrant, where $x < 0$ and $y > 0$ .
The graph of $f(x)$ lies in the:
$\boxed{\text{Second Quadrant}}$

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