Content

• Steps

One of the most important components of algebra is the concept of an inverse function. The inverse of the function is denoted as f ^ -1 (x) and is graphically represented as a reflection of the graph of the original function relative to the straight line y = x. In this article, we will show you how to find the inverse function.

Steps

1. 1 Make sure this function is bijective. Only bijective functions have inverse functions.
   • A function is bijective if it passes the vertical and horizontal line test. Draw a vertical line through the graph of the function and count the number of times that the line crosses the graph of the function. Then draw a horizontal line through the graph of the function and count the number of times that the line crosses the graph of the function. If each straight line intersects the graph of a function only once, then the function is bijective.
     • If the graph does not pass the vertical line test, then it is not specified by the function.
   • For an algebraic definition of the bijectivity of a function, substitute f (a) and f (b) into this function and determine whether the equality a = b holds. As an example, consider the function f (x) = 3x + 5.
     • f (a) = 3a + 5; f (b) = 3b + 5
     • 3a + 5 = 3b + 5
     • 3a = 3b
     • a = b
   • Thus, this function is bijective.
2. 2 In this function, swap "x" and "y". Remember that f (x) is a different spelling for "y".
   • "f (x)" or "y" is a function, and "x" is a variable. To find the inverse function, you need to swap the function and variable.
   • Example: Consider a function f (x) = (4x + 3) / (2x + 5), which is bijective. By swapping "x" and "y", you get x = (4y + 3) / (2y + 5).
3. 3 Find "y". Solve the new equation and find "y".
   • You may need algebraic tricks like multiplication of fractions or factoring to find the meaning of an expression and to simplify it.
   • Solution to our example:
     • x = (4y + 3) / (2y + 5)
     • x (2y + 5) = 4y + 3 - get rid of the fraction. To do this, multiply both sides of the equation by the denominator of the fraction (2y + 5).
     • 2xy + 5x = 4y + 3 - expand the brackets.
     • 2xy - 4y = 3 - 5x - Move all the terms with the variable (in this case, "y") to one side of the equation.
     • y (2x - 4) = 3 - 5x - place "y" outside the bracket.
     • y = (3 - 5x) / (2x - 4) - Divide both sides of the equation by (2x-4) to get your final answer.
4. 4 Replace "y" with f ^ -1 (x). This is the inverse function of the original function.
   • The final answer is f ^ -1 (x) = (3 - 5x) / (2x - 4). This is the inverse function for f (x) = (4x + 3) / (2x + 5).