Inverse functions

Inverse Functions

An inverse function essentially reverses the effect of the original function. If you have a function f(x) that takes an input x and produces an output y, the inverse function, denoted as f^-1(y), takes that output y and gives you back the original input x.

Definition of Inverse Functions

A function f is said to have an inverse if:

• f: A → B is a function that maps elements from set A to set B.
• There exists a function g: B → A such that for every element x in A, f(g(x)) = x and for every element y in B, g(f(y)) = y.

In simpler terms, if you apply the function f to an input x and then apply the inverse function f^-1 to the output, you should get back the original input x.

Finding Inverse Functions

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).

Example 1: Finding the Inverse of a Linear Function

Consider the function f(x) = 2x + 3.

1. Replace f(x) with y: y = 2x + 3.
2. Swap x and y: x = 2y + 3.
3. Solve for y:

x - 3 = 2y

y = (x - 3)/2

4. Replace y with f^-1(x): f^-1(x) = (x - 3)/2.

Thus, the inverse function of f(x) = 2x + 3 is f^-1(x) = (x - 3)/2.

Example 2: Finding the Inverse of a Quadratic Function

Now consider the function f(x) = x² + 1. Note that this function is not one-to-one, meaning it does not pass the horizontal line test. We can restrict the domain to x ≥ 0 to find an inverse.

1. Replace f(x) with y: y = x² + 1.
2. Swap x and y: x = y² + 1.
3. Solve for y:

x - 1 = y2

y = √(x - 1)

4. Replace y with f^-1(x): f^-1(x) = √(x - 1), for x ≥ 1.

Thus, the inverse function of f(x) = x² + 1, when restricted to x ≥ 0, is f^-1(x) = √(x - 1).

Graphing Inverse Functions

The graph of an inverse function is a reflection of the graph of the original function across the line y = x. This means that if you take any point (a, b) on the graph of the function f, the point (b, a) will be on the graph of the inverse function f^-1.

Example 3: Graphing a Function and Its Inverse

Consider the function f(x) = 2x + 3. Its inverse is f^-1(x) = (x - 3)/2.

To graph these functions:

• Graph f(x) = 2x + 3. This is a straight line with a slope of 2 and a y-intercept at (0, 3).
• Graph f^-1(x) = (x - 3)/2. This is also a straight line with a slope of 0.5 and a y-intercept at (3, 0).

Properties of Inverse Functions

Inverse functions have several important properties:

• If f and g are inverse functions, then f(g(x)) = x and g(f(x)) = x.
• The domain of f is the range of f^-1, and the range of f is the domain of f^-1.
• If f is increasing, then f^-1 is also increasing. If f is decreasing, then f^-1 is decreasing.

Check Your Understanding

• Find the inverse of the function f(x) = 3x - 4.
• Determine if the function f(x) = x² - 4 has an inverse. If yes, find it.
• Graph the function f(x) = x + 2 and its inverse. Identify key points.
• Explain why the function f(x) = x³ has an inverse.