Finding The Inverse Of F(x)=-3/4x+9 A Step-by-Step Guide

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In mathematics, the inverse of a function, denoted as $f^{-1}(x)$, essentially undoes the operation performed by the original function, $f(x)$. In simpler terms, if $f(a) = b$, then $f^{-1}(b) = a$. Finding the inverse of a function is a fundamental concept in algebra and calculus, with applications ranging from solving equations to understanding transformations of graphs. This guide will provide a clear, step-by-step approach to finding the inverse of a linear function, using the example of $f(x) = -\frac{3}{4}x + 9$. This specific function is a linear equation, representing a straight line when graphed. The process of finding its inverse involves algebraic manipulation to isolate the input variable, effectively reversing the function's operations. Understanding this process is crucial for students and anyone working with mathematical functions.

Step 1: Replace $f(x)$ with y

The initial step in finding the inverse function is to replace the function notation, $f(x)$, with the variable y. This substitution makes the equation easier to manipulate algebraically. For our example, $f(x) = -\frac3}{4}x + 9$, we replace $f(x)$ with y, resulting in the equation $y = -\frac{3{4}x + 9$. This transformation doesn't change the function itself; it simply provides a more convenient form for the subsequent steps. The y variable represents the output of the function for a given input x. By making this substitution, we set the stage for swapping the roles of x and y in the next step, which is the key to finding the inverse. This seemingly simple step is crucial for visually and conceptually understanding the reversal of the function's operation. It allows us to treat the equation in a more symmetrical manner, preparing it for the algebraic manipulations that will follow. Remember, the goal is to isolate x in terms of y, which will then allow us to express the inverse function.

Step 2: Swap x and y

This step is the heart of finding the inverse function. We interchange the roles of x and y in the equation. This swapping reflects the fundamental concept of an inverse function: it reverses the input and output of the original function. So, in our equation, $y = -\frac3}{4}x + 9$, we swap x and y to get $x = -\frac{3{4}y + 9$. This seemingly simple act has profound implications. It transforms the equation from expressing y in terms of x to expressing x in terms of y. This is precisely what we need to do to find the inverse. By swapping the variables, we are essentially looking at the function from the opposite perspective. Instead of asking, "What is the output (y) for a given input (x)?" we are now asking, "What input (y, which will become our $x$ in the inverse function) produces a given output (x, which will become our $y$ in the inverse function)?" This step highlights the symmetry between a function and its inverse. They are mirror images of each other across the line $y = x$. The swapping of variables is the algebraic manifestation of this geometric relationship. This step may seem abstract at first, but it is essential for understanding the core concept of inverse functions.

Step 3: Solve for y

Now that we've swapped x and y, our next task is to isolate y on one side of the equation. This involves using algebraic manipulations to undo the operations performed on y. In our example equation, $x = -\frac3}{4}y + 9$, we need to get y by itself. First, we subtract 9 from both sides of the equation $x - 9 = -\frac{34}y$. Next, to eliminate the fraction, we multiply both sides by -4/3 $-\frac{4{3}(x - 9) = y$. This is a crucial step in the process. Solving for y essentially reverses the original function's operations. We are performing the opposite operations in the reverse order to isolate y. This step often requires careful attention to the order of operations and the rules of algebra. It's where your algebraic skills will be put to the test. By isolating y, we are expressing y as a function of x, which is exactly what we need for the inverse function. The solution for y will be the algebraic expression that defines the inverse function. Make sure to carefully distribute and simplify the expression to obtain the final form of the inverse function. This may involve distributing the -4/3, combining like terms, and ensuring the expression is in its simplest form. The resulting equation will then be ready for the final step, where we replace y with the inverse function notation.

Step 4: Replace y with $f^{-1}(x)$

The final step in finding the inverse function is to replace the y we just solved for with the inverse function notation, $f^-1}(x)$. This notation explicitly denotes that we have found the inverse of the original function $f(x)$. In our example, after solving for y, we arrived at $y = -\frac{4}{3}(x - 9)$. Distributing the -4/3, we get $y = -\frac{4}{3}x + 12$. Now, we replace y with $f^{-1}(x)$ to express our final answer $f^{-1(x) = -\frac{4}{3}x + 12$. This notation clearly indicates that this function is the inverse of the original function, $f(x) = -\frac{3}{4}x + 9$. The notation $f^{-1}(x)$ is not to be interpreted as $1/f(x)$. It is a specific notation that represents the inverse function. This final step completes the process of finding the inverse. We have successfully manipulated the equation to isolate y and then expressed the result using the proper inverse function notation. The resulting function, $f^{-1}(x)$, when applied to the output of the original function, $f(x)$, will return the original input. This is the defining characteristic of an inverse function. By replacing y with $f^{-1}(x)$, we formally define the inverse function and make it clear that we have completed the process.

In summary, to find the inverse of the function $f(x)=-\frac{3}{4}x+9$, we follow these steps:

  1. Replace $f(x)$ with y: $y = -\frac{3}{4}x + 9$
  2. Swap x and y: $x = -\frac{3}{4}y + 9$
  3. Solve for y:

    • x−9=−34yx - 9 = -\frac{3}{4}y

    • −43(x−9)=y-\frac{4}{3}(x - 9) = y

    • y=−43x+12y = -\frac{4}{3}x + 12

  4. Replace y with $f^-1}(x)$ $f^{-1(x) = -\frac{4}{3}x + 12$

Therefore, the inverse of the function $f(x) = -\frac{3}{4}x + 9$ is $f^{-1}(x) = -\frac{4}{3}x + 12$. This step-by-step process can be applied to find the inverse of many different types of functions, especially linear functions. Understanding the concept of inverse functions and mastering the steps to find them is a valuable skill in mathematics.

To ensure that the calculated inverse function, $f^-1}(x)$, is indeed the correct inverse of the original function, $f(x)$, we can perform a verification step. This involves composing the function and its inverse in both directions $f(f^{-1(x))$ and $f^{-1}(f(x))$. If both compositions result in x, then we have successfully found the inverse function. This verification step is crucial because it confirms that the inverse function truly