Inverse Functions

Front

What is an inverse function?

Back

An inverse function reverses the operation of the original function. If the function f takes an input x and produces an output y, then the inverse function f^{-1} takes y as input and produces x as output.

Front

How do you find the inverse of a function algebraically?

Back

To find the inverse of a function, follow these steps: 1. Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with f^{-1}(x).

Front

What is the inverse of f(x) = -4x - 12?

Back

The inverse is f^{-1}(x) = 4x - 3.

Front

What are the coordinates of the inverse of the points (1, 2), (3, 8), (-3, 6)?

Back

The inverse coordinates are (2, 1), (8, 3), (6, -3).

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 across the line y = x.

Front

What is the relationship between a function and its inverse?

Back

A function and its inverse undo each other. For any x in the domain of f, f(f^{-1}(x)) = x and f^{-1}(f(x)) = x.

Front

What is a one-to-one function?

Back

A one-to-one function is a function where each output is produced by exactly one input, ensuring that the inverse is also a function.

Front

How can you determine if a function has an inverse?

Back

A function has an inverse if it is one-to-one, which can be tested using the horizontal line test: if any horizontal line intersects the graph more than once, the function does not have an inverse.

Front

What is the inverse of the function f(x) = 2x + 3?

Back

The inverse is f^{-1}(x) = \frac{x - 3}{2}.

Front

What is the inverse of the function f(x) = x^2 (for x >= 0)?

Back

The inverse is f^{-1}(x) = \sqrt{x}.