Find The Inverse Of $f(x) = -3x + 4$.A. $f^{-1}(x) = -\frac{1}{3}x - 4$B. $f^{-1}(x) = -\frac{1}{3}x + \frac{1}{4}$C. $f^{-1}(x) = -\frac{1}{3}x + \frac{4}{3}$D. $f^{-1}(x) = 3x - 4$

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Introduction

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 focus on finding the inverse of a linear function, specifically the function f(x)=−3x+4f(x) = -3x + 4. We will explore the steps involved in finding the inverse and provide a detailed solution to the problem.

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)=−3x+4f(x) = -3x + 4, the slope of the line is −3-3 and the y-intercept is 44.

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: Inverse functions can be used to solve systems of equations by substituting the inverse function into one of the equations.
  • Graphing functions: Inverse functions can be used to graph functions by reflecting the graph of the original function across the line y=xy = x.
  • Modeling real-world problems: Inverse functions can be used to model real-world problems, such as the motion of an object under the influence of gravity.

Step 1: Write the Function as y=f(x)y = f(x)

To find the inverse of a function, we need to start by writing the function as y=f(x)y = f(x). In this case, we have:

y=−3x+4y = -3x + 4

Step 2: Swap the Variables xx and yy

Next, we need to swap the variables xx and yy. This will give us:

x=−3y+4x = -3y + 4

Step 3: Solve for yy

Now, we need to solve for yy. To do this, we can isolate yy on one side of the equation. We can start by subtracting 44 from both sides:

x−4=−3yx - 4 = -3y

Step 4: Divide Both Sides by −3-3

Next, we need to divide both sides by −3-3 to solve for yy:

x−4−3=y\frac{x - 4}{-3} = y

Step 5: Simplify the Expression

Finally, we can simplify the expression by combining the terms on the left-hand side:

y=−13x+43y = -\frac{1}{3}x + \frac{4}{3}

Conclusion

In this article, we have found the inverse of the linear function f(x)=−3x+4f(x) = -3x + 4. The inverse function is given by:

f−1(x)=−13x+43f^{-1}(x) = -\frac{1}{3}x + \frac{4}{3}

We have also discussed the importance of finding the inverse of a function and provided a step-by-step guide on how to find the inverse of a linear function.

Answer

The correct answer is:

  • C. f−1(x)=−13x+43f^{-1}(x) = -\frac{1}{3}x + \frac{4}{3}

Discussion

  • What is the inverse of a function?
    • The inverse of a function is a function that reverses the operation of the original function.
  • Why is finding the inverse of a function important?
    • Finding the inverse of a function is essential in various mathematical applications, such as solving systems of equations, graphing functions, and modeling real-world problems.
  • How do you find the inverse of a linear function?
    • To find the inverse of a linear function, you need to start by writing the function as y=f(x)y = f(x). Then, you need to swap the variables xx and yy. Next, you need to solve for yy by isolating it on one side of the equation. Finally, you can simplify the expression to find the inverse function.

Example Problems

  • Find the inverse of the function f(x)=2x−1f(x) = 2x - 1
    • To find the inverse of the function f(x)=2x−1f(x) = 2x - 1, we need to start by writing the function as y=f(x)y = f(x). Then, we need to swap the variables xx and yy. Next, we need to solve for yy by isolating it on one side of the equation. Finally, we can simplify the expression to find the inverse function.
  • Find the inverse of the function f(x)=x+2f(x) = x + 2
    • To find the inverse of the function f(x)=x+2f(x) = x + 2, we need to start by writing the function as y=f(x)y = f(x). Then, we need to swap the variables xx and yy. Next, we need to solve for yy by isolating it on one side of the equation. Finally, we can simplify the expression to find the inverse function.

Practice Problems

  • Find the inverse of the function f(x)=−2x+3f(x) = -2x + 3
  • Find the inverse of the function f(x)=x−2f(x) = x - 2
  • Find the inverse of the function f(x)=3x+1f(x) = 3x + 1

Conclusion

In this article, we have found the inverse of the linear function f(x)=−3x+4f(x) = -3x + 4. We have also discussed the importance of finding the inverse of a function and provided a step-by-step guide on how to find the inverse of a linear function. We have also provided example problems and practice problems for readers to try.

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Introduction

In our previous article, we discussed the concept of inverse functions and provided a step-by-step guide on how to find the inverse of a linear function. In this article, we will answer some frequently asked questions about inverse functions.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of the original function. In other words, if we have a function f(x)f(x), then its inverse function f−1(x)f^{-1}(x) will "undo" the operation of f(x)f(x).

Q: Why is finding the inverse of a function important?

A: Finding the inverse of a function is essential in various mathematical applications, such as solving systems of equations, graphing functions, and modeling real-world problems.

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

A: To find the inverse of a function, you need to start by writing the function as y=f(x)y = f(x). Then, you need to swap the variables xx and yy. Next, you need to solve for yy by isolating it on one side of the equation. Finally, you can simplify the expression to find the inverse function.

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

A: The main difference between a function and its inverse is that the function and its inverse are reflections of each other across the line y=xy = x. In other words, if we have a function f(x)f(x), then its inverse function f−1(x)f^{-1}(x) will have the same graph as f(x)f(x), but reflected across the line y=xy = x.

Q: Can a function have more than one inverse?

A: No, a function cannot have more than one inverse. The inverse of a function is unique and is denoted by f−1(x)f^{-1}(x).

Q: How do you know if a function has an inverse?

A: A function has an inverse if and only if it is one-to-one, meaning that each value of xx corresponds to a unique value of yy. In other words, if a function is one-to-one, then it has an inverse.

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

A: The relationship between a function and its inverse is that the function and its inverse are inversely proportional. In other words, if we have a function f(x)f(x), then its inverse function f−1(x)f^{-1}(x) will have the property that f(f−1(x))=xf(f^{-1}(x)) = x and f−1(f(x))=xf^{-1}(f(x)) = x.

Q: Can a function have an inverse if it is not one-to-one?

A: No, a function cannot have an inverse if it is not one-to-one. If a function is not one-to-one, then it does not have an inverse.

Q: How do you find the inverse of a quadratic function?

A: To find the inverse of a quadratic function, you need to start by writing the function as y=f(x)y = f(x). Then, you need to swap the variables xx and yy. Next, you need to solve for yy by isolating it on one side of the equation. Finally, you can simplify the expression to find the inverse function.

Q: Can a function have an inverse if it is a polynomial of degree greater than 1?

A: No, a function cannot have an inverse if it is a polynomial of degree greater than 1. If a function is a polynomial of degree greater than 1, then it is not one-to-one and does not have an inverse.

Conclusion

In this article, we have answered some frequently asked questions about inverse functions. We have discussed the concept of inverse functions, how to find the inverse of a function, and the relationship between a function and its inverse. We have also provided examples and explanations to help readers understand the concept of inverse functions.

Practice Problems

  • Find the inverse of the function f(x)=2x−1f(x) = 2x - 1
  • Find the inverse of the function f(x)=x+2f(x) = x + 2
  • Find the inverse of the function f(x)=3x+1f(x) = 3x + 1

Example Problems

  • Find the inverse of the function f(x)=−2x+3f(x) = -2x + 3
  • Find the inverse of the function f(x)=x−2f(x) = x - 2
  • Find the inverse of the function f(x)=3x+1f(x) = 3x + 1

Discussion

  • What is the difference between a function and its inverse?
    • The main difference between a function and its inverse is that the function and its inverse are reflections of each other across the line y=xy = x.
  • Can a function have more than one inverse?
    • No, a function cannot have more than one inverse. The inverse of a function is unique and is denoted by f−1(x)f^{-1}(x).
  • How do you know if a function has an inverse?
    • A function has an inverse if and only if it is one-to-one, meaning that each value of xx corresponds to a unique value of yy.