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
False? Because -1 doesn't equal 1/3? Still don't get it
Is it false because -1 isn't equal to 1/3
the y-intercept of f(x) is an x-intercept of f^-1
Look at the example I did for you.
What is the y-intercept of f(x)?
What is the x-intercept of f^-1?
Are they the same?
If yes, true.
If not, false.
Oh, good one! I guess you could say it's a "reverse crossover" situation! But unfortunately, it's false. Although an inverse function undoes the effects of the original function, it doesn't necessarily mean that their intercepts will match up. The x-intercept of f inverse might not be the same as the y-intercept of f. They can dance to different beats, you know!
False.
To explain why this statement is false, let's break it down.
Firstly, let's clarify some terminology. The inverse function of f, denoted as f^(-1), is a function that undoes the operation of f. In other words, if you apply f followed by f^(-1) or f^(-1) followed by f, you will get back to the original input.
The y-intercept of a function is the point where the graph intersects the y-axis, which is represented as (0, f(0)). Similarly, an x-intercept is a point where the graph intersects the x-axis, represented as (x, 0).
Now, let's consider a scenario where the inverse function of f exists, and the graph of f has a y-intercept. Since the y-intercept is a point on the y-axis, it can be represented as (0, f(0)).
If the y-intercept of f were to be an x-intercept of f^(-1), it would mean that the point (0, f(0)) must lie on the x-axis. However, since the x-axis is where the y-coordinate is always 0, this would imply that f(0) = 0.
In general, there is no guarantee that the y-intercept of f will have a corresponding x-intercept on the inverse function f^(-1). The x-intercept of f^(-1) is related to the y-intercept of f and not necessarily the same; it depends on the specific nature of the two functions.
Therefore, the statement is false.
This exact question was answered yesterday.
Are you Sam too?
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?