Discovering the Inverse of a Function: Steps and Examples

Inverse Function

An inverse function is a function that undoes the actions of another function

An inverse function is a function that undoes the actions of another function. In other words, if we have a function f(x) that takes an input x and gives an output y, then the inverse function of f, denoted as f^(-1)(y) or simply f^(-1), takes the output y and gives back the input x.

To determine if a function has an inverse, we need to check if the function is one-to-one, meaning that each input has a unique output. One way to determine if a function is one-to-one is by examining its graph. If a graph passes both the horizontal line test and the vertical line test, then it is one-to-one and has an inverse function.

Finding the inverse of a function algebraically involves the following steps:

1. Start with the original function, let’s say f(x).

2. Replace f(x) with y.

3. Interchange the roles of x and y (meaning, replace x with y and y with x).

4. Solve the resulting equation for y in terms of x. This will give us the expression for the inverse function.

5. Replace y with f^(-1)(x) to express the inverse function as f^(-1)(x) = …

Let’s work through an example to illustrate these steps:

Example: Consider the function f(x) = 2x + 3. Find its inverse function.

1. Start with f(x) = 2x + 3.

2. Replace f(x) with y: y = 2x + 3.

3. Interchange x and y: x = 2y + 3.

4. Solve for y: x – 3 = 2y.

Divide both sides by 2: (x – 3)/2 = y.

5. 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.

To verify that a function and its inverse are indeed inverses, you can compose them. That is, if you plug in the output of a function into its inverse and get the original input, and vice versa, then they are inverses.

For example, if we plug f(x) = 2x + 3 into its inverse f^(-1)(x) = (x – 3)/2, we should get x:

f^(-1)(f(x)) = ((2x + 3) – 3)/2 = 2x/2 = x.

Similarly, if we plug f^(-1)(x) = (x – 3)/2 into the original function f(x) = 2x + 3, we should get x:

f(f^(-1)(x)) = 2((x – 3)/2) + 3 = x – 3 + 3 = x.

Thus, we have confirmed that f(x) = 2x + 3 and its inverse f^(-1)(x) = (x – 3)/2 are indeed inverses.

Remember, finding the inverse of a function is not always possible. The function must be one-to-one for an inverse to exist.

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