Finding The Inverse Function Of F(x) = (1/9)x + 2 A Step-by-Step Guide

In mathematics, particularly in algebra, understanding the concept of inverse functions is crucial. An inverse function essentially reverses the operation of the original function. If a function $f(x)$ takes an input $x$ and produces an output $y$, its inverse, denoted as $f^{-1}(x)$, takes $y$ as input and returns the original $x$. This article delves into the process of finding the inverse of a specific linear function, $f(x) = \frac{1}{9}x + 2$. We will explore the underlying principles and provide a step-by-step solution to determine the correct inverse function from the given options.

Understanding Inverse Functions

Before we dive into the specific problem, let's solidify our understanding of inverse functions. A function, in simple terms, is a rule that assigns each input value to exactly one output value. The inverse function, if it exists, undoes this operation. Not all functions have inverses; for a function to have an inverse, it must be one-to-one, meaning that each output value corresponds to only one input value. Graphically, this can be checked using the horizontal line test: if any horizontal line intersects the graph of the function at most once, then the function is one-to-one and has an inverse. Linear functions (except for horizontal lines) are always one-to-one and thus have inverses.

The inverse function is found by switching the roles of the input and output variables (typically $x$ and $y$) and then solving for the new output variable. This process essentially reverses the mapping defined by the original function. For example, if $f(a) = b$, then $f^{-1}(b) = a$. This fundamental relationship is key to understanding and finding inverse functions. It's also important to note the notation: $f^{-1}(x)$ represents the inverse function of $f(x)$, not the reciprocal of $f(x)$ (which would be written as $[f(x)]^{-1}$ or $\frac{1}{f(x)}$). The domain of the inverse function is the range of the original function, and vice versa. This reciprocal relationship between domains and ranges further emphasizes the reversing nature of inverse functions. When dealing with real-world applications, understanding the inverse function allows us to solve problems from a different perspective, often providing valuable insights and solutions that might not be immediately apparent from the original function alone.

Step-by-Step Solution for $f(x) = \frac{1}{9}x + 2$

Now, let's apply the concept of inverse functions to our specific problem: finding the inverse of $f(x) = \frac{1}{9}x + 2$. We'll follow a clear, step-by-step approach to ensure we arrive at the correct solution.

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

This is a simple notational change to make the subsequent steps clearer. We rewrite the function as: $y = \frac{1}{9}x + 2$.

Step 2: Swap $x$ and $y$

This is the crucial step that embodies the reversing action of finding the inverse. We interchange the input and output variables, resulting in: $x = \frac{1}{9}y + 2$.

Step 3: Solve for $y$

Our goal now is to isolate $y$ on one side of the equation. This involves using algebraic manipulations to undo the operations performed on $y$. First, we subtract 2 from both sides: $x - 2 = \frac{1}{9}y$. Next, to get $y$ by itself, we multiply both sides by 9: $9(x - 2) = y$.

Step 4: Simplify and express in inverse function notation

Distribute the 9 on the left side: $9x - 18 = y$. Finally, we replace $y$ with the inverse function notation, $f^{-1}(x)$, to clearly indicate that this is the inverse function: $f^{-1}(x) = 9x - 18$.

Therefore, the inverse of the function $f(x) = \frac{1}{9}x + 2$ is $f^{-1}(x) = 9x - 18$. Comparing this result to the given options, we find that the correct answer is B. $h(x) = 9x - 18$.

Detailed Explanation of Each Option

To further solidify our understanding, let's analyze why the other options are incorrect and highlight common mistakes made when finding inverse functions.

• A. $h(x) = 18x - 2$: This option seems to involve some manipulation of the original equation, but it doesn't correctly reverse the operations. The coefficient of $x$ is incorrect, and the constant term also doesn't follow the correct algebraic steps for finding the inverse.
• C. $h(x) = 9x + 18$: This option has the correct coefficient for $x$ (which is 9), but the sign of the constant term is incorrect. This likely stems from an error in distributing or simplifying during the solving-for-$y$ step.
• D. $h(x) = 18x + 2$: This option incorrectly manipulates both the coefficient of $x$ and the constant term. It doesn't reflect the proper steps for inverting the given linear function.

The most common mistake when finding inverse functions is often an error in the algebraic manipulation while solving for $y$. Remember to perform the operations in the correct order and to apply them to both sides of the equation to maintain equality. Another common error is forgetting to distribute correctly when simplifying expressions. Careful attention to detail is crucial in these algebraic manipulations.

Verifying the Inverse Function

To ensure that we have found the correct inverse function, we can use a simple verification method. Recall that if $f^{-1}(x)$ is indeed the inverse of $f(x)$, then the following should hold true:

• $f(f^{-1}(x)) = x$
• $f^{-1}(f(x)) = x$

Let's verify our solution, $f^{-1}(x) = 9x - 18$, using the first condition:

$f(f^{-1}(x)) = f(9x - 18) = \frac{1}{9}(9x - 18) + 2 = x - 2 + 2 = x$

Now, let's verify using the second condition:

$f^{-1}(f(x)) = f^{-1}(\frac{1}{9}x + 2) = 9(\frac{1}{9}x + 2) - 18 = x + 18 - 18 = x$

Since both conditions hold true, we can confidently confirm that $f^{-1}(x) = 9x - 18$ is indeed the correct inverse function. This verification step is a powerful tool to ensure the accuracy of your solution and can save you from errors in exams or problem-solving situations.

Importance and Applications of Inverse Functions

The concept of inverse functions extends beyond mere mathematical exercises; it has significant practical applications in various fields. Understanding inverse functions allows us to solve problems in reverse, providing a different perspective and often simplifying complex scenarios. For example, in cryptography, inverse functions play a vital role in encoding and decoding messages. The encryption process can be viewed as a function that transforms plaintext into ciphertext, and the decryption process, which recovers the original plaintext, is essentially the inverse function.

In economics, inverse functions can be used to analyze supply and demand curves. The demand curve expresses the quantity of a product consumers are willing to buy at a given price, while the inverse demand curve expresses the price consumers are willing to pay for a given quantity. Similarly, supply curves and their inverses provide insights into the relationship between price and quantity from the perspective of producers. In computer graphics, transformations like rotations and scaling can be represented by matrices, and the inverse matrices represent the inverse transformations, allowing us to undo these operations.

Furthermore, in calculus, the concept of inverse functions is crucial for understanding inverse trigonometric functions and inverse hyperbolic functions, which have wide applications in physics and engineering. The ability to find and work with inverse functions is a fundamental skill in mathematics and a valuable tool for solving real-world problems across diverse disciplines. By understanding the principles and techniques discussed in this article, you can confidently tackle inverse function problems and appreciate their broader significance.

Conclusion

In summary, finding the inverse of the function $f(x) = \frac{1}{9}x + 2$ involves a series of straightforward algebraic steps: replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and expressing the result in inverse function notation. The correct inverse function is $f^{-1}(x) = 9x - 18$, which corresponds to option B. By understanding the concept of inverse functions and practicing the steps involved in finding them, you can confidently solve similar problems and appreciate the wide range of applications of this fundamental mathematical concept. Remember to verify your solution whenever possible to ensure accuracy, and don't hesitate to revisit the steps and explanations if you encounter difficulties. Mastering inverse functions is a significant step towards building a strong foundation in mathematics and its applications.