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

#Introduction

In mathematics, the concept of an inverse function is fundamental, particularly in algebra and calculus. The inverse of a function, denoted as f⁻¹(x), essentially undoes the operation performed by the original function, f(x). Understanding how to find the inverse of a function is crucial for solving various mathematical problems and grasping more advanced concepts. In this article, we will delve into the process of finding the inverse of the function f(x) = (3/4)x - 9, a linear function, and explore the underlying principles. We will also address a multiple-choice question related to this function, providing a step-by-step solution and a detailed explanation to ensure clarity.

Understanding Inverse Functions

To truly grasp the concept, let's start with the basics of inverse functions. An inverse function, f⁻¹(x), exists for a function f(x) if and only if f(x) is a one-to-one function, meaning that it passes both the horizontal and vertical line tests. This implies that for every y-value, there is only one corresponding x-value, and vice versa. When we talk about finding the inverse, we are essentially trying to reverse the roles of input (x) and output (y).

The inverse function, denoted as f⁻¹(x), effectively reverses the operation performed by the original function, f(x). Imagine f(x) as a machine that takes an input, x, and produces an output, y. The inverse function, f⁻¹(x), acts as a reverse machine, taking the output, y, and returning the original input, x. This relationship is mathematically expressed as:

• f⁻¹(f(x)) = x
• f(f⁻¹(x)) = x

These equations highlight the fundamental property of inverse functions: when a function is composed with its inverse, the result is the original input, x. This property is crucial for verifying whether a function is indeed the inverse of another.

To illustrate this with our given function, if we find the correct inverse function, substituting it back into the original function (or vice versa) should yield x. This will be a key step in verifying our solution later.

Steps to Find the Inverse of a Function

Finding the inverse of a function involves a systematic approach. Here are the key steps to follow:

1. Replace f(x) with y: This simplifies the equation and makes it easier to manipulate.
2. Swap x and y: This is the crucial step in finding the inverse, as it reverses the roles of input and output.
3. Solve for y: Isolate y on one side of the equation. This will give you the inverse function in terms of x.
4. Replace y with f⁻¹(x): This is the standard notation for the inverse function.

Understanding these steps is crucial for finding the inverse of any function, not just the one given. We'll apply these steps methodically to f(x) = (3/4)x - 9 to arrive at the correct inverse function.

Solving for the Inverse of f(x) = (3/4)x - 9

Now, let's apply these steps to our function, f(x) = (3/4)x - 9. This linear function represents a straight line with a slope of 3/4 and a y-intercept of -9. Finding its inverse will give us another linear function that reverses this relationship.

Step 1: Replace f(x) with y

First, we replace f(x) with y in the equation:

y = (3/4)x - 9

This substitution is a simple notational change, but it sets the stage for the next crucial step.

Step 2: Swap x and y

Next, we swap x and y:

x = (3/4)y - 9

This step is the heart of finding the inverse function. By swapping x and y, we are essentially reversing the input and output roles, which is the fundamental idea behind inverse functions.

Step 3: Solve for y

Now, we need to isolate y in the equation. This involves a few algebraic manipulations. First, we add 9 to both sides of the equation:

x + 9 = (3/4)y

Next, to get rid of the fraction, we multiply both sides by 4/3:

(4/3)(x + 9) = y

Distributing the 4/3 on the left side, we get:

(4/3)x + (4/3)(9) = y

(4/3)x + 12 = y

So, we have now solved for y in terms of x.

Step 4: Replace y with f⁻¹(x)

Finally, we replace y with f⁻¹(x) to denote the inverse function:

f⁻¹(x) = (4/3)x + 12

Therefore, the inverse of f(x) = (3/4)x - 9 is f⁻¹(x) = (4/3)x + 12.

This result matches option A in the multiple-choice question, which we will address next.

Multiple Choice Question and Answer

Now let's consider the multiple-choice question related to this function:

Question: The inverse of f(x) = (3/4)x - 9 is:

A. f⁻¹(x) = (4/3)x + 12

B. f⁻¹(x) = (4/3)x - 12

C. f⁻¹(x) = (4/3)x + 9

D. f⁻¹(x) = (3/4)x + 9

Solution

As we found in the previous section, the inverse of f(x) = (3/4)x - 9 is:

f⁻¹(x) = (4/3)x + 12

Therefore, the correct answer is A. f⁻¹(x) = (4/3)x + 12.

Verification

To verify our answer, we can compose the function with its inverse and check if we get x. Let's find f(f⁻¹(x)):

f(f⁻¹(x)) = f((4/3)x + 12)

= (3/4)((4/3)x + 12) - 9

= (3/4)(4/3)x + (3/4)(12) - 9

= x + 9 - 9

= x

Since f(f⁻¹(x)) = x, our answer is indeed correct. This verification step is a crucial check to ensure that the calculated inverse function is accurate.

Common Mistakes and How to Avoid Them

When finding inverse functions, there are several common mistakes that students often make. Understanding these mistakes and how to avoid them can significantly improve accuracy.

Common Mistakes

1. Incorrectly swapping x and y: The most crucial step in finding the inverse is swapping x and y. A common mistake is forgetting to do this or doing it incorrectly.
2. Algebraic errors while solving for y: Isolating y involves algebraic manipulations, and errors in these steps can lead to an incorrect inverse function. Common errors include incorrect distribution, sign errors, and incorrect order of operations.
3. Forgetting to replace y with f⁻¹(x): After solving for y, it's important to replace it with f⁻¹(x) to denote the inverse function correctly.
4. Not verifying the answer: It's always a good practice to verify the inverse function by composing it with the original function. Forgetting this step can lead to accepting an incorrect answer.

How to Avoid Mistakes

1. Double-check the swap: Ensure that you have correctly swapped x and y before proceeding with the algebraic manipulations.
2. Be careful with algebra: Take your time and be meticulous with each step. Pay attention to signs, distribution, and the order of operations. Write out each step clearly to minimize errors.
3. Use correct notation: Always replace y with f⁻¹(x) to clearly indicate that you have found the inverse function.
4. Verify your answer: Always verify your answer by composing the function with its inverse. If the result is not x, there is an error somewhere in your calculations.

By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your ability to find inverse functions accurately.

Real-World Applications of Inverse Functions

While inverse functions might seem like a purely theoretical concept, they have numerous applications in real-world scenarios. Understanding these applications can help to appreciate the practical significance of inverse functions.

Applications

1. Cryptography: In cryptography, inverse functions are used to encrypt and decrypt messages. The encryption process transforms a message into an unreadable form, and the inverse function is used to decrypt it back to its original form.
2. Computer Graphics: Inverse functions are used in computer graphics to perform transformations such as rotations, scaling, and translations. The inverse transformations are used to undo these operations.
3. Economics: In economics, inverse functions can be used to model supply and demand. For example, the demand function gives the quantity demanded as a function of price, while the inverse demand function gives the price as a function of quantity demanded.
4. Science and Engineering: Inverse functions are used in various scientific and engineering applications, such as converting between different units of measurement, solving equations, and modeling physical phenomena.

Examples

• Temperature Conversion: Converting between Celsius and Fahrenheit involves inverse functions. The formula to convert Celsius to Fahrenheit is F = (9/5)C + 32, and the inverse function to convert Fahrenheit to Celsius is C = (5/9)(F - 32).
• Financial Calculations: In finance, inverse functions can be used to calculate the principal amount needed to achieve a certain future value, given an interest rate and time period.
• Medical Dosage: In medicine, inverse functions can be used to determine the dosage of a drug needed to achieve a certain concentration in the bloodstream.

These examples illustrate the diverse applications of inverse functions in various fields. By understanding inverse functions, we can solve problems and model real-world phenomena more effectively.

Conclusion

In conclusion, understanding how to find the inverse of a function is a fundamental concept in mathematics with wide-ranging applications. In this article, we have demonstrated the step-by-step process of finding the inverse of the function f(x) = (3/4)x - 9, which is f⁻¹(x) = (4/3)x + 12. We have also addressed a multiple-choice question related to this function, providing a clear solution and verification.

We discussed the importance of inverse functions, the steps involved in finding them, and common mistakes to avoid. Furthermore, we explored real-world applications of inverse functions in cryptography, computer graphics, economics, science, and engineering.

By mastering the concept of inverse functions, you will not only enhance your mathematical skills but also gain a valuable tool for solving problems in various fields. The ability to find and apply inverse functions is crucial for further studies in mathematics and its applications.

Remember to practice finding the inverses of various functions to solidify your understanding. With practice and a clear understanding of the steps involved, you can confidently tackle inverse function problems and appreciate their significance in mathematics and beyond.