True or False?

If the inverse function of f exists, and the graph of f has a y-intercept, the y-intercept of f is an x-intercept of f^-1

Try an example to see if this is true.

f(x) = 3x - 1
So, f(x) has y-intercept = -1

To find inverse of f(x),
y = 3x - 1
x = 1/3 y + 1/3
y = 1/3 x + 1/3
f^-1 = 1/3 x + 1/3

To find x-intercept,
y = 1/3 x + 1/3
0 = 1/3 x + 1/3
-1/3 = 1/3 x
x = -1

So, f^-1 has an x-intercept = -1.

So, True or False?

YUT

true

The statement is false.

To understand why, let's break down the concepts involved.

First, let's clarify what it means for a function to have a y-intercept and an x-intercept.

- A y-intercept is a point in the Cartesian coordinate plane where a function intersects or touches the y-axis. It has coordinates (0, b), where b represents the y-coordinate of the point of intersection.

- An x-intercept, on the other hand, is a point where a function intersects or touches the x-axis. It has coordinates (a, 0), where a represents the x-coordinate of the point of intersection.

Now, let's consider the inverse of a function. The inverse function, denoted as f^(-1), is a function that "reverses" the effects of the original function f. In other words, if (x, y) is a point on the graph of f, then (y, x) will be a point on the graph of f^(-1).

Based on these definitions, we can see that the y-intercept of f, which is (0, b), does not necessarily have any relation to the x-intercept of f^(-1), which is (a, 0). The x-coordinate of the y-intercept of f is always 0, while the y-coordinate of the x-intercept of f^(-1) is also 0. However, there is no guarantee that these two points will share the same x-coordinate.

Therefore, the y-intercept of f is not an x-intercept of f^(-1) in general. So, the statement is false.