Inverse Function
An inverse function is a function that undoes the action of another function
An inverse function is a function that undoes the action of another function. In other words, if you apply a function to a number, and then apply its inverse function to the result, you should get back the original number.
To find the inverse of a function, you need to follow a few steps:
1. Start with the function, let’s call it f(x).
2. Replace f(x) with y.
3. Swap the roles of x and y in the equation. This means replacing x with y and y with x.
4. Solve the new equation for y.
5. Replace y with f^(-1)(x), which represents the inverse function.
Let’s go through an example to illustrate this process:
Consider the function f(x) = 2x + 3.
1. Replace f(x) with y: y = 2x + 3.
2. Swap the roles of x and y: x = 2y + 3.
3. Solve for y: x – 3 = 2y. Dividing by 2, we get y = (x – 3)/2.
4. Replace y with f^(-1)(x): f^(-1)(x) = (x – 3)/2.
So the inverse function of f(x) = 2x + 3 is f^(-1)(x) = (x – 3)/2.
To verify if two functions are inverses of each other, you can plug one into the other and see if they cancel out to give you the input. For example, let’s plug f(x) into f^(-1)(x):
f^(-1)(f(x)) = [(2x + 3) – 3]/2 = 2x/2 = x.
As expected, the result is x, which confirms that f(x) and f^(-1)(x) are indeed inverses of each other.
It’s important to note that not all functions have inverses. For a function to have an inverse, it must be one-to-one, meaning that each input corresponds to a unique output. If a function fails the horizontal line test, it does not have an inverse.
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