Find The Inverse Function Of F ( X ) = 4 X − 12 F(x) = 4x - 12 F ( X ) = 4 X − 12 . F − 1 ( X ) = □ X + □ F^{-1}(x) = \square X + \square F − 1 ( X ) = □ X + □

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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 f^(-1)(x) maps the output f(x) back to the input x. In this article, we will learn how to find the inverse function of a linear equation, specifically the function f(x) = 4x - 12.

What is a Linear Equation?

A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. In other words, a linear equation is an equation of the form ax + b = c, where a, b, and c are constants. The graph of a linear equation is a straight line.

The Function f(x) = 4x - 12

The function f(x) = 4x - 12 is a linear equation. To find the inverse function of this equation, we need to follow a series of steps.

Step 1: Write the Equation in Function Notation

The first step in finding the inverse function is to write the equation in function notation. In function notation, we write the equation as f(x) = 4x - 12.

Step 2: Switch the x and y Variables

The next step is to switch the x and y variables. In other words, we replace x with y and y with x. This gives us the equation x = 4y - 12.

Step 3: Solve the Equation for y

Now, we need to solve the equation for y. To do this, we add 12 to both sides of the equation, which gives us x + 12 = 4y. Then, we divide both sides of the equation by 4, which gives us y = (x + 12) / 4.

Step 4: Write the Inverse Function

The final step is to write the inverse function. Since we have solved the equation for y, we can write the inverse function as f^(-1)(x) = (x + 12) / 4.

The Inverse Function of f(x) = 4x - 12

So, the inverse function of f(x) = 4x - 12 is f^(-1)(x) = (x + 12) / 4.

Example

Let's say we want to find the inverse of the function f(x) = 4x - 12 when x = 3. To do this, we plug x = 3 into the inverse function f^(-1)(x) = (x + 12) / 4, which gives us f^(-1)(3) = (3 + 12) / 4 = 15 / 4 = 3.75.

Conclusion

In this article, we learned how to find the inverse function of a linear equation, specifically the function f(x) = 4x - 12. We followed a series of steps, including writing the equation in function notation, switching the x and y variables, solving the equation for y, and writing the inverse function. We also provided an example of how to use the inverse function to find the value of a function when the input is given.

Key Takeaways

  • The inverse function of a linear equation is a function that reverses the operation of the original function.
  • To find the inverse function of a linear equation, we need to follow a series of steps, including writing the equation in function notation, switching the x and y variables, solving the equation for y, and writing the inverse function.
  • The inverse function of f(x) = 4x - 12 is f^(-1)(x) = (x + 12) / 4.

Further Reading

If you want to learn more about inverse functions, I recommend checking out the following resources:

  • Khan Academy: Inverse Functions
  • Math Is Fun: Inverse Functions
  • Wolfram MathWorld: Inverse Function

References