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, swap the x and y in the equation and solve for y.

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 is the line of reflection for inverse functions?

Back

The line of reflection for inverse functions is y = x.

Front

If f(x) = 2x + 3, what is f⁻¹(x)?

Back

f⁻¹(x) = (x - 3)/2.

Front

What points does f⁻¹(x) go through if f(x) goes through (2, 5)?

Back

f⁻¹(x) goes through (5, 2).

Front

Are the functions f(x) = x² and g(x) = √x inverses of each other?

Back

No, because f(x) = x² is not one-to-one; it does not pass the horizontal line test.

Front

What is 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 is not one-to-one.

Front

If f(x) = 3x - 1, what is the inverse function f⁻¹(x)?

Back

f⁻¹(x) = (x + 1)/3.

Front

What is the relationship between a function and its inverse?

Back

If f(x) and f⁻¹(x) are inverses, then f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.