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.

Front

What is the inverse of the function f(x) = 3x - 5?

Back

f^-1(x) = (x + 5) / 3.

Front

What does it mean for two functions to be inverses of each other?

Back

Two functions are inverses if applying one function followed by the other returns the original input. That is, f(f^-1(x)) = x and f^-1(f(x)) = x.

Front

Can a function have more than one inverse?

Back

No, a function can have at most one inverse. If a function is not one-to-one, it does not have an inverse.

Front

What is the graphical representation of inverse functions?

Back

The graphs of inverse functions are reflections of each other across the line y = x.

Front

How can you determine if two functions are inverses using composition?

Back

To check if f and g are inverses, compute f(g(x)) and g(f(x)). If both equal x for all x in the domain, they are inverses.

Front

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

Back

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

Front

What is the significance of the horizontal line test?

Back

The horizontal line test determines if a function is one-to-one. If any horizontal line intersects the graph more than once, the function does not have an inverse.

Front

What is the inverse of a constant function?

Back

A constant function does not have an inverse because it is not one-to-one.