Inverse Functions

Front

What is an inverse function?

Back

An inverse function reverses the operation of the original function. If f(x) takes an input x and produces an output y, then the inverse function f^-1(y) takes y back to x.

Front

How do you find the inverse of a function algebraically?

Back

To find the inverse of a function, replace f(x) with y, swap x and y, and then solve for y. The resulting equation is the inverse function.

Front

What is the graphical representation of an inverse function?

Back

The graph of an inverse function is a reflection of the original function's graph over the line y = x.

Front

What does it mean if f(g(x)) = x?

Back

It means that f and g are inverse functions of each other, but this is only true if both functions are defined for the same domain.

Front

What is the significance of the line y = x in relation to inverse functions?

Back

The line y = x serves as the line of reflection for the graphs of a function and its inverse.

Front

If f(1) = 3, what is f^-1(3)?

Back

f^-1(3) = 1, because the inverse function reverses the input-output relationship.

Front

What is the relationship between the domain of a function and the range of its inverse?

Back

The domain of the original function becomes the range of the inverse function, and vice versa.

Front

How do you determine if a function has an inverse?

Back

A function has an inverse if it is one-to-one, meaning that it passes the horizontal line test (no horizontal line intersects the graph more than once).

Front

What is the inverse of f(x) = 2x + 5?

Back

f^-1(x) = (x - 5)/2.

Front

If f(x) = x², does it have an inverse?

Back

No, f(x) = x² is not one-to-one over the real numbers, so it does not have an inverse.