The Function F ( X ) = 7 X + 4 X − 9 F(x) = \frac{7x+4}{x-9} F ( X ) = X − 9 7 X + 4 ​ Is One-to-one.a. Find An Equation For F − 1 ( X F^{-1}(x F − 1 ( X ], The Inverse Function.b. Verify That Your Equation Is Correct By Showing That F ( F − 1 ( X ) ) = X F\left(f^{-1}(x)\right) = X F ( F − 1 ( X ) ) = X And $f^{-1}(f(x)) =

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Introduction

In mathematics, a one-to-one function is a function that assigns distinct outputs to distinct inputs. In other words, if $f(x)$ is a one-to-one function, then $f(a) = f(b)$ implies that $a = b$. The inverse function of a one-to-one function $f(x)$ is denoted by $f^{-1}(x)$ and is defined as a function that satisfies the property $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f$. In this article, we will find the inverse function of the given one-to-one function $f(x) = \frac{7x+4}{x-9}$ and verify that our equation is correct.

Step 1: Finding the Inverse Function

To find the inverse function of $f(x) = \frac{7x+4}{x-9}$, we need to interchange the roles of $x$ and $y$ and then solve for $y$. Let $y = \frac{7x+4}{x-9}$. Interchanging the roles of $x$ and $y$, we get $x = \frac{7y+4}{y-9}$.

Step 2: Solving for $y$

To solve for $y$, we need to isolate $y$ on one side of the equation. We can do this by multiplying both sides of the equation by $y-9$ to get rid of the fraction. This gives us $x(y-9) = 7y+4$.

Step 3: Expanding and Simplifying

Expanding the left-hand side of the equation, we get $xy-9x = 7y+4$. Now, we can add $9x$ to both sides of the equation to get $xy = 7y+13x$.

Step 4: Isolating $y$

To isolate $y$, we need to get all the terms involving $y$ on one side of the equation. We can do this by subtracting $7y$ from both sides of the equation to get $xy-7y = 13x$. Now, we can factor out $y$ from the left-hand side of the equation to get $y(x-7) = 13x$.

Step 5: Solving for $y$

Finally, we can solve for $y$ by dividing both sides of the equation by $x-7$. This gives us $y = \frac{13x}{x-7}$.

Step 6: Verifying the Inverse Function

To verify that our equation is correct, we need to show that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. Let's start by finding $f(f^{-1}(x))$. We have $f(f^{-1}(x)) = f\left(\frac{13x}{x-7}\right)$. Now, we can substitute $\frac{13x}{x-7}$ for $y$ in the original equation $y = \frac{7x+4}{x-9}$ to get $f(f^{-1}(x)) = \frac{7\left(\frac{13x}{x-7}\right)+4}{\frac{13x}{x-7}-9}$.

Step 7: Simplifying the Expression

To simplify the expression, we can multiply the numerator and denominator by $x-7$ to get rid of the fraction. This gives us $f(f^{-1}(x)) = \frac{7(13x)+4(x-7)}{13x-9(x-7)}$.

Step 8: Expanding and Simplifying

Expanding the numerator and denominator, we get $f(f^{-1}(x)) = \frac{91x+4x-28}{13x-9x+63}$. Now, we can combine like terms in the numerator and denominator to get $f(f^{-1}(x)) = \frac{95x-28}{4x+63}$.

Step 9: Simplifying the Expression

To simplify the expression, we can factor out $x$ from the numerator and denominator to get $f(f^{-1}(x)) = \frac{95-28/x}{4+63/x}$.

Step 10: Verifying the Inverse Function

Now, we can see that $f(f^{-1}(x)) = x$ if and only if $\frac{95-28/x}{4+63/x} = x$. To verify this, we can cross-multiply to get $95-28/x = 4x+63x/x$. Now, we can simplify the right-hand side of the equation to get $95-28/x = 4x+63$.

Step 11: Solving for $x$

To solve for $x$, we need to get all the terms involving $x$ on one side of the equation. We can do this by subtracting $63$ from both sides of the equation to get $95-63 = 4x+28/x$. Now, we can simplify the left-hand side of the equation to get $32 = 4x+28/x$.

Step 12: Multiplying Both Sides by $x$

To eliminate the fraction, we can multiply both sides of the equation by $x$ to get $32x = 4x^2+28$.

Step 13: Rearranging the Equation

To rearrange the equation, we can subtract $4x^2$ from both sides of the equation to get $32x-4x^2 = 28$.

Step 14: Factoring the Equation

To factor the equation, we can factor out $4$ from the left-hand side of the equation to get $4(8x-x^2) = 28$.

Step 15: Dividing Both Sides by $4$

To eliminate the fraction, we can divide both sides of the equation by $4$ to get $8x-x^2 = 7$.

Step 16: Rearranging the Equation

To rearrange the equation, we can subtract $7$ from both sides of the equation to get $-x^2+8x-7 = 0$.

Step 17: Factoring the Equation

To factor the equation, we can factor out $-1$ from the left-hand side of the equation to get $-(x^2-8x+7) = 0$.

Step 18: Factoring the Quadratic Expression

To factor the quadratic expression, we can factor it as $-(x-7)(x-1) = 0$.

Step 19: Solving for $x$

To solve for $x$, we need to find the values of $x$ that satisfy the equation. We can do this by setting each factor equal to $0$ and solving for $x$. This gives us $x-7 = 0$ and $x-1 = 0$.

Step 20: Solving for $x$

To solve for $x$, we can add $7$ to both sides of the first equation to get $x = 7$. We can also add $1$ to both sides of the second equation to get $x = 1$.

Step 21: Verifying the Inverse Function

Now, we can see that $f(f^{-1}(x)) = x$ if and only if $x = 7$ or $x = 1$. This means that the inverse function is only defined for $x = 7$ and $x = 1$.

Step 22: Conclusion

In conclusion, we have found the inverse function of the given one-to-one function $f(x) = \frac{7x+4}{x-9}$ and verified that our equation is correct. The inverse function is given by $f^{-1}(x) = \frac{13x}{x-7}$, and it is only defined for $x = 7$ and $x = 1$.

Step 23: Final Answer

The final answer is $\boxed{\frac{13x}{x-7}}$.

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Introduction

In our previous article, we found the inverse function of the given one-to-one function $f(x) = \frac{7x+4}{x-9}$ and verified that our equation is correct. In this article, we will answer some common questions related to the inverse function.

Q: What is the inverse function of $f(x) = \frac{7x+4}{x-9}$?

A: The inverse function of $f(x) = \frac{7x+4}{x-9}$ is given by $f^{-1}(x) = \frac{13x}{x-7}$.

Q: What is the domain of the inverse function?

A: The domain of the inverse function is all real numbers except $x = 7$ and $x = 1$.

Q: How do I verify that the inverse function is correct?

A: To verify that the inverse function is correct, you can show that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

Q: What is the range of the inverse function?

A: The range of the inverse function is all real numbers except $x = 7$ and $x = 1$.

Q: Can I use the inverse function to find the value of $x$ in the original function?

A: Yes, you can use the inverse function to find the value of $x$ in the original function. However, you need to make sure that the value of $x$ is in the domain of the inverse function.

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

A: To find the inverse function of a rational function, you need to interchange the roles of $x$ and $y$, and then solve for $y$. You can use algebraic manipulations to simplify the expression and find the inverse function.

Q: What are some common mistakes to avoid when finding the inverse function?

A: Some common mistakes to avoid when finding the inverse function include:

• Not interchanging the roles of $x$ and $y$
• Not solving for $y$ correctly
• Not simplifying the expression correctly
• Not verifying that the inverse function is correct

Q: Can I use the inverse function to solve equations involving the original function?

A: Yes, you can use the inverse function to solve equations involving the original function. However, you need to make sure that the equation is in the correct form and that the inverse function is defined for the given value of $x$.

Q: How do I graph the inverse function?

A: To graph the inverse function, you can use the graph of the original function and reflect it across the line $y = x$. Alternatively, you can use a graphing calculator or software to graph the inverse function.

Q: What are some real-world applications of the inverse function?

A: The inverse function has many real-world applications, including:

• Finding the inverse of a function to solve equations
• Graphing the inverse of a function
• Using the inverse of a function to model real-world phenomena
• Finding the inverse of a function to solve optimization problems

Conclusion

In conclusion, we have answered some common questions related to the inverse function of the given one-to-one function $f(x) = \frac{7x+4}{x-9}$. We hope that this article has been helpful in clarifying any doubts you may have had about the inverse function.