Inverse Equations

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 as input and produces x as output.

Front

How do you verify if two functions are inverses of each other?

Back

To verify if two functions f(x) and g(x) are inverses, you need to show that f(g(x)) = x and g(f(x)) = x for all x in the domain.

Front

What is the inverse of the function f(x) = 4x - 9?

Back

The inverse function is f^{-1}(x) = (x + 9)/4.

Front

What is the significance of the horizontal line test in relation to inverse functions?

Back

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

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 inverse of the function f(x) = (x - 2)^2?

Back

The inverse function is f^{-1}(x) = √x + 2, defined for x ≥ 0.

Front

What does it mean for a function to be one-to-one?

Back

A function is one-to-one if it assigns a unique output for every unique input, meaning no two different inputs produce the same output.

Front

What is the inverse of the function g(x) = 1/4 x + 9/4?

Back

The inverse function is g^{-1}(x) = 4x - 9.

Front

What is the relationship between a function and its inverse in terms of their graphs?

Back

The graph of a function and its inverse are symmetric with respect to the line y = x.

Front

What is the inverse of the function f(x) = 1/2 (x - 3)^3 - 7?

Back

The inverse function is f^{-1}(x) = 3 + (2x + 14)^(1/3).