Finding The Inverse Function Of F(x) = (3-x)/5

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In mathematics, the inverse of a function essentially undoes what the original function does. If you input a value into a function and then input the result into its inverse, you should get back your original value. In this article, we will explore the process of finding the inverse of a function, using the example function $f(x) = \frac{3-x}{5}$. We will go through the steps meticulously, ensuring a clear understanding of the underlying concepts. This will help not only solve the given problem but also equip you with the skills to tackle similar problems in the future. Understanding inverse functions is crucial in many areas of mathematics, including calculus and algebra, so mastering this concept is a valuable investment in your mathematical journey.

Understanding Inverse Functions

Before we dive into the specific problem, let's first clarify what an inverse function is. Imagine a function as a machine that takes an input, performs some operations on it, and produces an output. The inverse function is like a machine that reverses this process. If the original function takes x and gives y, the inverse function takes y and gives x. Mathematically, if $f(x) = y$, then the inverse function, denoted as $f^{-1}(x)$, satisfies $f^{-1}(y) = x$.

To find the inverse of a function, we typically follow these steps:

  1. Replace $f(x)$ with y. This makes the equation easier to manipulate.
  2. Swap x and y. This is the crucial step that reverses the roles of input and output.
  3. Solve the equation for y. This isolates the inverse function.
  4. Replace y with $f^{-1}(x)$. This gives the inverse function in standard notation.

Understanding these steps is key to successfully finding the inverse of any function. By practicing these steps with different functions, you'll develop a strong intuition for how inverse functions work and be able to apply the concept in various mathematical contexts. The ability to find inverse functions is not just a theoretical exercise; it has practical applications in fields like cryptography, where reversing a function is essential for decoding messages.

Step-by-Step Solution

Now, let's apply these steps to find the inverse of the given function, $f(x) = \frac{3-x}{5}$. This function takes an input x, subtracts it from 3, and then divides the result by 5. Our goal is to find a function that reverses these operations, taking the output and returning the original input.

Step 1 Replace $f(x)$ with y

Our first step is to replace $f(x)$ with y. This simplifies the notation and makes the algebraic manipulations easier to follow. So, we rewrite the function as:

y=3−x5y = \frac{3-x}{5}

This equation now represents the same relationship as the original function, but in a form that is more conducive to finding the inverse. The variable y now represents the output of the function for a given input x. This substitution is a common technique in mathematics to make equations more manageable and easier to work with.

Step 2 Swap x and y

This is the core step in finding the inverse. We swap x and y to reverse the roles of input and output. This reflects the fundamental concept of an inverse function, which is to undo the original function's operation. After swapping, our equation becomes:

x=3−y5x = \frac{3-y}{5}

Notice how x now stands where y used to be, and y stands where x used to be. This swap effectively sets up the equation to solve for the inverse function. By solving for y in this equation, we will find the expression that represents the inverse function, $f^{-1}(x)$.

Step 3 Solve the equation for y

Now we need to isolate y in the equation $x = \frac{3-y}{5}$. This involves a series of algebraic manipulations to get y by itself on one side of the equation. The goal is to undo the operations that are being applied to y, working backwards from the order of operations.

First, we multiply both sides of the equation by 5 to eliminate the fraction:

5x=3−y5x = 3 - y

Next, we want to isolate the term with y. We can do this by adding y to both sides and subtracting 5x from both sides:

5x+y=35x + y = 3

y=3−5xy = 3 - 5x

Now we have successfully isolated y. This equation expresses y in terms of x, which is exactly what we need for the inverse function. The algebraic steps we took here are crucial for correctly finding the inverse. Each step must be performed carefully to maintain the equality and ensure that the final expression for y is accurate.

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

The final step is to replace y with the notation for the inverse function, $f^{-1}(x)$. This gives us the inverse function in standard notation, which is how we typically express inverse functions. Replacing y with $f^{-1}(x)$ in our equation gives:

f−1(x)=3−5xf^{-1}(x) = 3 - 5x

This is the inverse function of the original function, $f(x) = \frac{3-x}{5}$. The notation $f^{-1}(x)$ clearly indicates that this function is the inverse of f(x). This final step is important for clarity and consistency in mathematical notation.

Checking the Answer

To ensure that our answer is correct, we can verify it by composing the original function with its inverse. If $f^{-1}(x)$ is indeed the inverse of f(x), then the following should be true:

f(f−1(x))=xf(f^{-1}(x)) = x

and

f−1(f(x))=xf^{-1}(f(x)) = x

Let's check the first composition:

f(f−1(x))=f(3−5x)=3−(3−5x)5=3−3+5x5=5x5=xf(f^{-1}(x)) = f(3 - 5x) = \frac{3 - (3 - 5x)}{5} = \frac{3 - 3 + 5x}{5} = \frac{5x}{5} = x

Now let's check the second composition:

f−1(f(x))=f−1(3−x5)=3−5(3−x5)=3−(3−x)=3−3+x=xf^{-1}(f(x)) = f^{-1}(\frac{3 - x}{5}) = 3 - 5(\frac{3 - x}{5}) = 3 - (3 - x) = 3 - 3 + x = x

Both compositions result in x, which confirms that we have found the correct inverse function. This verification step is a crucial part of the process, as it helps to catch any errors that may have occurred during the algebraic manipulations. By checking our answer, we can be confident that our solution is correct.

Conclusion

Therefore, the inverse of the function $f(x) = \frac{3-x}{5}$ is:

D. $f^{-1}(x) = 3 - 5x$

We arrived at this answer by following a step-by-step process: replacing $f(x)$ with y, swapping x and y, solving for y, and then replacing y with $f^{-1}(x)$. We also verified our answer by composing the original function with its inverse and confirming that the result was x. Understanding how to find the inverse of a function is a valuable skill in mathematics, and this example provides a clear illustration of the process. By practicing these steps with different functions, you can build your proficiency and confidence in working with inverse functions.

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