If $g(x)$ Is The Inverse Of $f(x)$ And $f(x) = 4x + 12$, What Is $ G ( X ) G(x) G ( X ) [/tex]?A. $g(x) = 12x + 4$ B. $g(x) = \frac{1}{4}x - 12$ C. $ G ( X ) = X − 3 G(x) = X - 3 G ( X ) = X − 3 [/tex] D.

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Introduction

In mathematics, an inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function g(x) maps the output f(x) back to the input x. In this article, we will explore how to find the inverse of a linear function, specifically the inverse of the function f(x) = 4x + 12.

What is an Inverse Function?

An inverse function is a function that undoes the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function g(x) maps the output f(x) back to the input x. The inverse function is denoted by g(x) and is read as "g of x".

Properties of Inverse Functions

Inverse functions have several important properties that make them useful in mathematics. Some of the key properties of inverse functions include:

  • One-to-One Correspondence: An inverse function is a one-to-one correspondence between the input and output values of the original function.
  • Symmetry: The graph of an inverse function is symmetric to the graph of the original function about the line y = x.
  • Reversibility: An inverse function reverses the operation of the original function.

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: Switch the x and y variables in the original function.
  2. Solve for y: Solve the resulting equation for y.
  3. Interchange x and y: Interchange the x and y variables in the resulting equation.

Example: Finding the Inverse of f(x) = 4x + 12

Let's find the inverse of the function f(x) = 4x + 12.

Step 1: Switch x and y

Switch the x and y variables in the original function:

y = 4x + 12

Step 2: Solve for y

Solve the resulting equation for y:

x = 4y + 12

Subtract 12 from both sides:

x - 12 = 4y

Divide both sides by 4:

(x - 12) / 4 = y

Step 3: Interchange x and y

Interchange the x and y variables in the resulting equation:

y = (x - 12) / 4

This is the inverse function g(x).

Conclusion

In this article, we have explored how to find the inverse of a linear function, specifically the inverse of the function f(x) = 4x + 12. We have followed the steps of switching x and y, solving for y, and interchanging x and y to find the inverse function g(x) = (x - 12) / 4. We have also discussed the properties of inverse functions, including one-to-one correspondence, symmetry, and reversibility.

Answer

The correct answer is:

B. g(x) = (x - 12) / 4

Introduction

Inverse functions are a fundamental concept in mathematics, and understanding how to find the inverse of a function is crucial for solving problems in algebra, calculus, and other areas of mathematics. In this article, we will provide a Q&A guide to help you understand the concept of inverse functions and how to find the inverse of a function.

Q: What is an inverse function?

A: An inverse function is a function that reverses the operation of another function. In other words, if we have a function f(x) that maps an input x to an output f(x), then the inverse function g(x) maps the output f(x) back to the input x.

Q: What are the properties of inverse functions?

A: Inverse functions have several important properties, including:

  • One-to-One Correspondence: An inverse function is a one-to-one correspondence between the input and output values of the original function.
  • Symmetry: The graph of an inverse function is symmetric to the graph of the original function about the line y = x.
  • Reversibility: An inverse function reverses the operation of the original function.

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

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

  1. Switch x and y: Switch the x and y variables in the original function.
  2. Solve for y: Solve the resulting equation for y.
  3. Interchange x and y: Interchange the x and y variables in the resulting equation.

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

A: If the original function is a linear function, then the inverse function will also be a linear function. To find the inverse of a linear function, you can use the following formula:

g(x) = (x - b) / a

where a and b are the coefficients of the original function.

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

A: If the original function is a quadratic function, then the inverse function will not be a quadratic function. To find the inverse of a quadratic function, you need to use the quadratic formula to solve for the roots of the equation.

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

A: Yes, you can use a calculator to find the inverse of a function. Most graphing calculators have a built-in function to find the inverse of a function.

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

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

  • Not switching x and y: Make sure to switch the x and y variables in the original function.
  • Not solving for y: Make sure to solve the resulting equation for y.
  • Not interchanging x and y: Make sure to interchange the x and y variables in the resulting equation.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of inverse functions and how to find the inverse of a function. We discussed the properties of inverse functions, how to find the inverse of a linear function, and how to avoid common mistakes when finding the inverse of a function.

Common Inverse Functions

Here are some common inverse functions:

  • Inverse of f(x) = x + 3: g(x) = x - 3
  • Inverse of f(x) = 2x - 1: g(x) = (x + 1) / 2
  • Inverse of f(x) = x^2 + 1: g(x) = ±√(x - 1)

Practice Problems

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

  • Find the inverse of f(x) = 3x + 2.
  • Find the inverse of f(x) = x^2 - 4.
  • Find the inverse of f(x) = 2x - 5.

Answer Key

Here are the answers to the practice problems:

  • g(x) = (x - 2) / 3
  • g(x) = ±√(x + 4)
  • g(x) = (x + 5) / 2