Identify The Inverse Of The Function $f(x) = -6x + 7$.A. $f^{-1}(x) = \frac{x-7}{6}$ B. $f^{-1}(x) = \frac{-x+7}{6}$ C. $f^{-1}(x) = -\frac{x}{6} - 7$ D. $f^{-1}(x) = \frac{x}{6} - 7$

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In mathematics, the concept of inverse functions is crucial in understanding the relationship between two functions. The inverse of a function essentially reverses the operation of the original function. In this article, we will explore how to find the inverse of a linear function, specifically the function f(x)=โˆ’6x+7f(x) = -6x + 7.

What is a Linear Function?

A linear function is a polynomial function of degree one, which means it has the form f(x)=ax+bf(x) = ax + b, where aa and bb are constants. The graph of a linear function is a straight line. In the case of the function f(x)=โˆ’6x+7f(x) = -6x + 7, the slope of the line is โˆ’6-6 and the y-intercept is 77.

Why Find the Inverse of a Function?

Finding the inverse of a function is essential in various mathematical applications, such as solving systems of equations, graphing functions, and modeling real-world phenomena. The inverse of a function helps us to:

  • Solve equations involving the function
  • Graph the function and its inverse
  • Understand the relationship between the function and its inverse

Step-by-Step Guide to Finding the Inverse of a Linear Function

To find the inverse of a linear function, we need to follow these steps:

  1. Switch x and y: Swap the x and y variables in the original function. This means that the new function will have the form x=f(y)x = f(y).
  2. Solve for y: Solve the new function for y in terms of x.
  3. Interchange x and y: Interchange the x and y variables in the resulting function.

Finding the Inverse of f(x)=โˆ’6x+7f(x) = -6x + 7

Let's apply the steps above to find the inverse of the function f(x)=โˆ’6x+7f(x) = -6x + 7.

Step 1: Switch x and y

Switching x and y in the original function, we get:

x=โˆ’6y+7x = -6y + 7

Step 2: Solve for y

To solve for y, we need to isolate y on one side of the equation. We can do this by subtracting 7 from both sides and then dividing both sides by -6.

xโˆ’7=โˆ’6yx - 7 = -6y

xโˆ’7โˆ’6=y\frac{x - 7}{-6} = y

Step 3: Interchange x and y

Interchanging x and y in the resulting function, we get:

fโˆ’1(x)=xโˆ’7โˆ’6f^{-1}(x) = \frac{x - 7}{-6}

Simplifying the expression, we get:

fโˆ’1(x)=โˆ’x+76f^{-1}(x) = \frac{-x + 7}{6}

Conclusion

In this article, we have learned how to find the inverse of a linear function, specifically the function f(x)=โˆ’6x+7f(x) = -6x + 7. We have followed the steps of switching x and y, solving for y, and interchanging x and y to find the inverse function. The inverse function is fโˆ’1(x)=โˆ’x+76f^{-1}(x) = \frac{-x + 7}{6}.

Answer

The correct answer is:

A. fโˆ’1(x)=xโˆ’76f^{-1}(x) = \frac{x-7}{6}

However, this is not the correct answer. The correct answer is:

In the previous article, we explored how to find the inverse of a linear function, specifically the function f(x)=โˆ’6x+7f(x) = -6x + 7. In this article, we will answer some frequently asked questions about finding the inverse of a linear function.

Q: What is the inverse of a linear function?

A: The inverse of a linear function is a function that reverses the operation of the original function. In other words, if the original function takes an input x and produces an output y, the inverse function takes the output y and produces the input x.

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

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

  1. Switch x and y in the original function.
  2. Solve for y in terms of x.
  3. Interchange x and y in the resulting function.

Q: What if the original function is in the form f(x)=ax+bf(x) = ax + b?

A: If the original function is in the form f(x)=ax+bf(x) = ax + b, you can find the inverse by following the same steps as above. However, you need to be careful when solving for y, as the equation may not be linear.

Q: Can I use a calculator to find the inverse of a linear function?

A: Yes, you can use a calculator to find the inverse of a linear function. However, it's always a good idea to check your work by hand to make sure the calculator is giving you the correct answer.

Q: What if the original function is a quadratic function?

A: If the original function is a quadratic function, you cannot find its inverse using the same steps as above. Quadratic functions have two solutions for y, which means they do not have an inverse.

Q: Can I find the inverse of a linear function with a negative slope?

A: Yes, you can find the inverse of a linear function with a negative slope. The steps for finding the inverse are the same as above, but you need to be careful when solving for y, as the equation may not be linear.

Q: What is the relationship between the original function and its inverse?

A: The original function and its inverse are related in the following way:

  • The original function takes an input x and produces an output y.
  • The inverse function takes the output y and produces the input x.

Q: Can I graph the original function and its inverse?

A: Yes, you can graph the original function and its inverse. The graph of the original function and its inverse are reflections of each other across the line y = x.

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

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

Conclusion

In this article, we have answered some frequently asked questions about finding the inverse of a linear function. We have covered topics such as the definition of an inverse function, how to find the inverse of a linear function, and the relationship between the original function and its inverse.

Common Mistakes to Avoid

When finding the inverse of a linear function, there are several common mistakes to avoid:

  • Switching x and y incorrectly
  • Solving for y incorrectly
  • Interchanging x and y incorrectly
  • Not checking the work by hand

Tips and Tricks

When finding the inverse of a linear function, here are some tips and tricks to keep in mind:

  • Make sure to switch x and y correctly
  • Solve for y carefully
  • Interchange x and y carefully
  • Check the work by hand to make sure the calculator is giving you the correct answer

Practice Problems

Here are some practice problems to help you practice finding the inverse of a linear function:

  1. Find the inverse of the function f(x)=2x+3f(x) = 2x + 3.
  2. Find the inverse of the function f(x)=โˆ’4x+2f(x) = -4x + 2.
  3. Find the inverse of the function f(x)=xโˆ’1f(x) = x - 1.

Answer Key

  1. fโˆ’1(x)=xโˆ’32f^{-1}(x) = \frac{x - 3}{2}
  2. fโˆ’1(x)=โˆ’x+24f^{-1}(x) = \frac{-x + 2}{4}
  3. fโˆ’1(x)=x+1f^{-1}(x) = x + 1