How to find the inverse of a function

General
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how to find the inverse of a function

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How to find the inverse of a function

Answer:

To find the inverse of a function, the following steps can be followed:

1. Understand the function:
The first step is to understand the given function. Analyze the relationship between the independent variable (input) and the dependent variable (output) of the function.

2. Interchange the variables:
To find the inverse of a function, we need to interchange the independent and dependent variables. This means that the input of the original function becomes the output of the inverse function, and the output of the original function becomes the input of the inverse function.

3. Solve for the inverse function:
After interchanging the variables, we need to solve the resulting equation for the inverse function. This involves isolating the independent variable (which is now the output of the inverse function) on one side of the equation.

4. Verify the inverse function:
Once you have obtained the inverse function, it’s important to verify that it is indeed the inverse of the original function. You can do this by substituting values into both the original function and the inverse function to ensure that they “undo” each other.

Example:
Let’s consider the function f(x) = 2x + 3. To find the inverse function, we follow the steps:

  1. Understand the function:
    The function f(x) = 2x + 3 is a linear function, where the input x is multiplied by 2 and then 3 is added to the result.

  2. Interchange the variables:
    We interchange the independent variable x and the dependent variable y, so we have y = 2x + 3.

  3. Solve for the inverse function:
    To solve for the inverse function, we need to isolate x on one side of the equation:
    y = 2x + 3
    y - 3 = 2x
    \frac{y - 3}{2} = x
    Therefore, the inverse function is f^{-1}(x) = \frac{x - 3}{2}.

  4. Verify the inverse function:
    To verify that f^{-1}(x) = \frac{x - 3}{2} is the inverse of f(x) = 2x + 3, we can substitute values and check that they “undo” each other.
    For example, if x = 5, then f(x) = 2(5) + 3 = 13. Substituting y = 13 into the inverse function, we get f^{-1}(13) = \frac{13 - 3}{2} = 5, which is the original input.

The process of finding the inverse of a function involves interchanging the independent and dependent variables, and then solving the resulting equation for the inverse function. This allows us to “undo” the original function and find the input value given the output value.