Carlie is studying a function, *f*(*x*), that has a y-intercept at (0, 8). She immediately concludes that its inverse, *f*−1(*x*), must have its yy-intercept at (8, 0). She feels good about this at first, but the longer she thinks about it, the less sure she is.

Is Carlie’s conclusion correct?

A.Carlie is incorrect. (8, 0) is the *x*-intercept of *f−1(x)*.

B.Carlie is incorrect. The y-intercept of f−1(*x*) has to be the inverse, which is (0,18)(0,18).

C.Carlie is incorrect. The *y*-intercept of *f*−1(*x*) is (0, −8).

D.Carlie is correct. Points match between inverse functions by switching the coordinates.

Carlie's conclusion is correct.

When finding the inverse of a function, the roles of the input (\(x\)) and output (\(y\)) are switched. This means that if the original function \(f(x)\) has a point \((a, b)\) on its graph, then the inverse function \(f^{-1}(x)\) will have the point \((b, a)\) on its graph. 

Given that \(f(x)\) has a y-intercept at \((0, 8)\), the corresponding point on the inverse function \(f^{-1}(x)\) will indeed be \((8, 0)\). Since \((8, 0)\) is a point on the inverse function, it is the y-intercept of \(f^{-1}(x)\).

So, **Carlie's conclusion is correct**.

**Answer: D. Carlie is correct. Points match between inverse functions by switching the coordinates.**

Would you like more details or have any questions? Here are 5 related questions:

1. How do you find the inverse of a function algebraically?
2. What conditions must a function satisfy to have an inverse?
3. How do the graphs of a function and its inverse relate to each other?
4. Can a function and its inverse have the same y-intercept?
5. What happens if a function has no y-intercept—does its inverse have an x-intercept?

**Tip:** When switching between a function and its inverse, always swap the \(x\) and \(y\) coordinates of any points to find corresponding points on the graphs.

Carlie is studying a function, *f*(x), that has a y-intercept at (0, 8). She immediately concludes that its inverse, *f*−1(x), must have its y-intercept at (8, 0). She feels good about this at first, but the longer she thinks about it, the less sure she is. Is Carlie’s conclusion correct?

Explore the relationship between a function and its inverse through a detailed analysis of Carlie's problem. Learn how the coordinates of points on a function's graph relate to those on its inverse. Suitable for students in Grades 9-11.

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