A fundamental part in learning algebra consists in learning how to find the inverse of a function f (x), which is denoted by f -1 (x) and visually it is represented by the original function reflected with respect to the line y = x. This article will show you how to find the inverse of a function.
Steps
Step 1. Make sure the function is "one to one", ie one-to-one
Only these functions have an inverse.
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A function is one-to-one if it passes the vertical and horizontal line test. Draw a vertical line across the entire graph of the function and count the number of times the line cuts the function. Then draw a horizontal line across the entire graph of the function and count the number of times this line takes the function. If each line cuts the function only once, the function is one-to-one.
If a graph does not pass the vertical line test, it is not even a function
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To algebraically determine if the function is one-to-one, setting f (a) = f (b), we must find that a = b. For example, let's take f (x) = 3 x + 5.
- f (a) = 3a + 5; f (b) = 3b + 5
- 3a + 5 = 3b + 5
- 3a = 3b
- a = b
- F (x) is thus one-to-one.
Step 2. Given a function, replace the x's with the y's:
remember that f (x) stands for "y".
- In a function, "f" or "y" represents the output and "x" represents the input. To find the inverse of a function, the inputs and outputs are inverted.
- Example: let's take f (x) = (4x + 3) / (2x + 5), which is one-to-one. By switching x to y, we get x = (4y + 3) / (2y + 5).
Step 3. Solve for the new "y"
You will need to modify the expressions to resolve with respect to y or to find the new operations that need to be performed on the input to get the inverse as the output.
- This can be difficult depending on your expression. You may need to use algebraic tricks like cross multiplication or factoring to evaluate the expression and simplify it.
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In our example, we will follow the steps below to isolate y:
- We are starting with x = (4y + 3) / (2y + 5)
- x (2y + 5) = 4y + 3 - Multiply both sides by (2y + 5)
- 2xy + 5x = 4y + 3 - Multiply by x
- 2xy - 4y = 3-5 x - Put all y terms aside
- y (2x - 4) = 3 - 5x - Collect the y
- y = (x 3-5) / (2 x - 4) - Divide to get your answer
Find the Inverse of a Function Step 4 Step 4. Replace the new "y" with f -1 (x).
This is the equation for the inverse of the original function.
Our final answer is f -1 (x) = (3-5 x) / (2x - 4). This is the inverse function of f (x) = (4x + 3) / (2x + 5).
