Inverse Of A Function

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Function Notation
Functions
Graphs Of Functions
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What is a one-to-one function?
If a function is defined so that each range element is used only once, then it is a one-to-one function.

• Every y-value is only paired with one x-value.
• A horizontal line test helps determine graphically whether a function is one-to-one. If the horizontal line intersects the graph of a function at more than one point then it is not one-to-one.
• Only one-to-one functions have inverse functions.

What is the Inverse of a Function?
We have learned that a function f maps x to f(x).
The inverse of f is a function which maps f(x) to x in reverse.
The inverse of the function f is denoted by f^-1.

The inverse of a function is found by interchanging its range and domain. The domain of F becomes the range of the inverse and the range of F becomes the domain of the inverse of F. The inverse of a function is not always a function and should be checked by the definition of a function. A function only has an inverse if it is one-to-one.

How to find the inverse of a function?
The steps involved in getting the inverse of a function are:
Step 1: Determine if the function is one to one.
Step 2: Interchange the x and y variables. This new function is the inverse function
Step 3: If the result is an equation, solve the equation for y.
Step 4: Replace y by f^-1(x), symbolizing the inverse function or the inverse of f.

We can perform this procedure on any function, but the resulting inverse will only be another function if the original function is a one-to-one function.

The following examples illustrates these steps.

Example:
Find the inverse of each of the following functions:

1. f = {(1,2), (-2,3), (5,-2)}
2. y = x³ + 2
3. \(y = \frac{2}{{x - 4}}\)

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How to find the inverse of a function, step by step examples
Find the Inverse of a Square Root Function with Domain and Range

Example:
Let \(f(x) = \sqrt {2x - 1} - 3\). Determine the domain and range. Then find f^-1(x).

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How to find the inverse of a function or show that the inverse does not exists

1. Replace f(x) with y.
   
2. Switch x’s and y’s.
   
3. Solve for y.
   
4. Replace ‘y’ with f^-1(x)

Examples:
Find the inverse of
\(f(x) = \sqrt {x + 4} - 3\)
\(y = \frac{{5x - 3}}{{2x + 1}}\)

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Find the inverse of the function if one exists
\(f(x) = \frac{{8 - {x^2}}}{5}\)
This video shows algebraically that the above function does not have an inverse.

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Find the Inverse Function of an Exponential Function

Example:
f(x) = 3e^2x

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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