A function f has an inverse function. If the graph of f^-1 lies in quadrant I, in which quadrant does the graph of f lie?
Understand the Problem
The question is asking us to determine the quadrant in which the graph of the function f lies based on the information that its inverse function f^-1 is in Quadrant I. We know that if the inverse of a function is in Quadrant I, then the function itself must be in Quadrant III, as inverse functions correspond to reflections across the line y = x.
Answer
Quadrant III
Answer for screen readers
The function $f$ lies in Quadrant III.
Steps to Solve
- Identify the properties of inverse functions
Inverse functions reflect across the line $y = x$. If a function $f(x)$ has an inverse $f^{-1}(x)$, the coordinates of the points on $f(x)$ and $f^{-1}(x)$ are related by swapping the x and y values.
- Determine the implication of Quadrant I
If the inverse function $f^{-1}(x)$ is located in Quadrant I, it means both x and y coordinates of points on this graph are positive. Thus, for every point $(x, y)$ on $f^{-1}(x)$, the corresponding point $(y, x)$ on $f(x)$ must have $y > 0$ and $x > 0$.
- Conclusion about the original function's location
Given that $f^{-1}$ is in Quadrant I, the original function $f(x)$, represented by points $(y, x)$, will be located where $y < 0$ and $x < 0$, which indicates that $f(x)$ is in Quadrant III.
The function $f$ lies in Quadrant III.
More Information
Quadrant III is where both x and y coordinates are negative. This is consistent with the property that the inverse of a function will have its coordinates swapped, leading to $f^{-1}(x)$ being in Quadrant I when $f(x)$ is in Quadrant III.
Tips
- Confusing the relationships of quadrants; remember that inverse functions reflect across the line $y = x$.
- Not applying the correct signs for the coordinates when determining which quadrant a function lies in.
