Arrange The Equations In The Correct Sequence To Find The Inverse Of $f(x) = Y = \frac{z-1}{33-x}$.1. $x(33-y) = Y-4$2. $33x - Xy = Y + 4$3. $33x + 4 = Y + Xy$4. $33x + 4 = Y(1 + X$\]5. $x =

Feb 26, 2025 by ADMIN 191 views

**Finding the Inverse of a Function: A Step-by-Step Guide**

Introduction

Finding the inverse of a function is an essential concept in mathematics, particularly in algebra and calculus. It involves reversing the function to obtain a new function that undoes the original function's operation. In this article, we will explore how to find the inverse of a given function, specifically the function $f(x) = y = \frac{z-1}{33-x}$ . We will arrange the equations in the correct sequence to find the inverse of this function.

Understanding the Function

Before we begin, let's understand the given function. The function is defined as $f(x) = y = \frac{z-1}{33-x}$ . This function takes an input $x$ and produces an output $y$ . To find the inverse of this function, we need to swap the roles of $x$ and $y$ and solve for $y$ in terms of $x$ .

Step 1: Swapping the Roles of x and y

To find the inverse of the function, we start by swapping the roles of $x$ and $y$ . This means that we replace $x$ with $y$ and $y$ with $x$ in the original function. The new function becomes $f(x) = y = \frac{x-1}{33-y}$ .

Step 2: Simplifying the Function

Now that we have swapped the roles of $x$ and $y$ , we need to simplify the function. We can start by multiplying both sides of the equation by $33-y$ to eliminate the fraction. This gives us $x(33-y) = x-1$ .

Step 3: Rearranging the Equation

Next, we need to rearrange the equation to isolate the term $y$ . We can do this by subtracting $x$ from both sides of the equation, which gives us $x(33-y) - x = -1$ . We can then factor out $x$ from the left-hand side of the equation, which gives us $x(33-y-1) = -1$ .

Step 4: Simplifying the Equation

Now that we have factored out $x$ , we can simplify the equation further. We can start by dividing both sides of the equation by $x$ , which gives us $33-y-1 = \frac{-1}{x}$ .

Step 5: Rearranging the Equation

Next, we need to rearrange the equation to isolate the term $y$ . We can do this by subtracting $33-1$ from both sides of the equation, which gives us $-y = \frac{-1}{x} - 32$ . We can then multiply both sides of the equation by $-1$ to get rid of the negative sign, which gives us $y = \frac{1}{x} + 32$ .

Step 6: Simplifying the Equation

Now that we have isolated the term $y$ , we can simplify the equation further. We can start by combining the terms on the right-hand side of the equation, which gives us $y = \frac{1+32x}{x}$ .

Step 7: Simplifying the Equation

Finally, we can simplify the equation further by combining the terms in the numerator. This gives us $y = \frac{32x+1}{x}$ .

Conclusion

In this article, we have explored how to find the inverse of a given function, specifically the function $f(x) = y = \frac{z-1}{33-x}$ . We have arranged the equations in the correct sequence to find the inverse of this function. The final answer is $y = \frac{32x+1}{x}$ .

Discussion

The process of finding the inverse of a function involves reversing the function to obtain a new function that undoes the original function's operation. This is an essential concept in mathematics, particularly in algebra and calculus. In this article, we have shown how to find the inverse of a given function by swapping the roles of $x$ and $y$ , simplifying the function, rearranging the equation, and simplifying the equation further.

Final Answer

The final answer is $y = \frac{32x+1}{x}$ .

References

[1] "Inverse Functions" by Math Open Reference
[2] "Finding the Inverse of a Function" by Khan Academy
[3] "Inverse Functions" by Wolfram MathWorld

Additional Resources

[1] "Inverse Functions" by MIT OpenCourseWare
[2] "Finding the Inverse of a Function" by Purplemath
[3] "Inverse Functions" by IXL Math

FAQs

Q: What is the inverse of a function? A: The inverse of a function is a new function that undoes the original function's operation.
Q: How do I find the inverse of a function? A: To find the inverse of a function, you need to swap the roles of $x$ and $y$ , simplify the function, rearrange the equation, and simplify the equation further.
Q: What is the final answer for the inverse of the function $f(x) = y = \frac{z-1}{33-x}$ ? A: The final answer is $y = \frac{32x+1}{x}$ .

Introduction

Inverse functions are a fundamental concept in mathematics, particularly in algebra and calculus. In this article, we will provide a comprehensive Q&A guide on inverse functions, covering various topics and concepts related to this subject.

Q: What is an inverse function?

A: An inverse function is a new function that undoes the original function's operation. In other words, if a function f(x) maps an input x to an output y, then the inverse function f^(-1)(x) maps the output y back to the input x.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to follow these steps:

Swap the roles of x and y in the original function.
Simplify the function by combining like terms and eliminating fractions.
Rearrange the equation to isolate the term y.
Simplify the equation further by combining terms and eliminating negative signs.

Q: What is the difference between a function and its inverse?

A: A function and its inverse are two different functions that are related to each other. The function f(x) maps an input x to an output y, while the inverse function f^(-1)(x) maps the output y back to the input x.

Q: How do I determine if a function has an inverse?

A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. You can determine if a function is one-to-one by checking if it passes the horizontal line test.

Q: What is the horizontal line test?

A: The horizontal line test is a method used to determine if a function is one-to-one. If a horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one and does not have an inverse.

Q: How do I graph the inverse of a function?

A: To graph the inverse of a function, you need to follow these steps:

Graph the original function.
Reflect the graph of the original function across the line y = x.
The resulting graph is the graph of the inverse function.

Q: What is the domain and range of an inverse function?

A: The domain and range of an inverse function are the same as the range and domain of the original function, respectively.

Q: How do I find the inverse of a composite function?

A: To find the inverse of a composite function, you need to follow these steps:

Find the inverse of each individual function in the composite function.
Combine the inverses to form the inverse of the composite function.

Q: What is the difference between an inverse function and a reciprocal function?

A: An inverse function and a reciprocal function are two different functions that are related to each other. The inverse function f^(-1)(x) maps the output y back to the input x, while the reciprocal function 1/f(x) maps the input x to the output 1/y.

Q: How do I find the inverse of a rational function?

A: To find the inverse of a rational function, you need to follow these steps:

Swap the roles of x and y in the original function.
Simplify the function by combining like terms and eliminating fractions.
Rearrange the equation to isolate the term y.
Simplify the equation further by combining terms and eliminating negative signs.

Q: What is the final answer for the inverse of the function f(x) = y = (z-1)/(33-x)?

A: The final answer is y = (32x+1)/x.

Conclusion

In this article, we have provided a comprehensive Q&A guide on inverse functions, covering various topics and concepts related to this subject. We hope that this guide has been helpful in understanding the concept of inverse functions and how to find the inverse of a function.

References

[1] "Inverse Functions" by Math Open Reference
[2] "Finding the Inverse of a Function" by Khan Academy
[3] "Inverse Functions" by Wolfram MathWorld

Additional Resources

[1] "Inverse Functions" by MIT OpenCourseWare
[2] "Finding the Inverse of a Function" by Purplemath
[3] "Inverse Functions" by IXL Math

FAQs

Q: What is the inverse of a function? A: The inverse of a function is a new function that undoes the original function's operation.
Q: How do I find the inverse of a function? A: To find the inverse of a function, you need to swap the roles of x and y, simplify the function, rearrange the equation, and simplify the equation further.
Q: What is the final answer for the inverse of the function f(x) = y = (z-1)/(33-x)? A: The final answer is y = (32x+1)/x.