If $f(x)=\frac{1}{9} X-2$, What Is $f^{-1}(x)$?A. $f^{-1}(x)=9 X+18$ B. $f^{-1}(x)=\frac{1}{9} X+2$ C. \$f^{-1}(x)=9 X+2$[/tex\] D. $f^{-1}(x)=-2 X+\frac{1}{9}$

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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 explore how to find the inverse function of a linear function.

What is a Linear Function?

A linear function is a function of the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept. The graph of a linear function is a straight line. In the given problem, we have a linear function f(x) = (1/9)x - 2.

Finding the Inverse Function

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

  1. Swap the x and y variables: We start by swapping the x and y variables in the original function. This gives us x = (1/9)y - 2.
  2. Solve for y: Next, we need to solve for y in the equation x = (1/9)y - 2. To do this, we add 2 to both sides of the equation, which gives us x + 2 = (1/9)y.
  3. Multiply both sides by 9: To isolate y, we multiply both sides of the equation by 9, which gives us 9(x + 2) = y.
  4. Simplify the equation: Finally, we simplify the equation 9(x + 2) = y to get y = 9x + 18.

Conclusion

In conclusion, the inverse function of f(x) = (1/9)x - 2 is f^(-1)(x) = 9x + 18. This means that if we plug in a value of x into the original function, we will get a value of y, and if we plug in the value of y into the inverse function, we will get back the original value of x.

Example

Let's use an example to illustrate this concept. Suppose we have the function f(x) = (1/9)x - 2, and we want to find the inverse function. We follow the steps outlined above:

  1. Swap the x and y variables: We start by swapping the x and y variables in the original function. This gives us x = (1/9)y - 2.
  2. Solve for y: Next, we need to solve for y in the equation x = (1/9)y - 2. To do this, we add 2 to both sides of the equation, which gives us x + 2 = (1/9)y.
  3. Multiply both sides by 9: To isolate y, we multiply both sides of the equation by 9, which gives us 9(x + 2) = y.
  4. Simplify the equation: Finally, we simplify the equation 9(x + 2) = y to get y = 9x + 18.

Answer

The inverse function of f(x) = (1/9)x - 2 is f^(-1)(x) = 9x + 18.

Comparison with Options

Let's compare our answer with the options given:

  • Option A: f^(-1)(x) = 9x + 18
  • Option B: f^(-1)(x) = (1/9)x + 2
  • Option C: f^(-1)(x) = 9x + 2
  • Option D: f^(-1)(x) = -2x + (1/9)

Our answer matches option A.

Conclusion

In conclusion, the inverse function of f(x) = (1/9)x - 2 is f^(-1)(x) = 9x + 18. This means that if we plug in a value of x into the original function, we will get a value of y, and if we plug in the value of y into the inverse function, we will get back the original value of x.

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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 f^(-1)(x) maps the output f(x) back to the input x.

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

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

  1. Swap the x and y variables: Swap the x and y variables in the original function.
  2. Solve for y: Solve for y in the equation x = (1/9)y - 2.
  3. Multiply both sides by 9: Multiply both sides of the equation by 9 to isolate y.
  4. Simplify the equation: Simplify the equation to get the inverse function.

Q: What is the inverse function of f(x) = (1/9)x - 2?

A: The inverse function of f(x) = (1/9)x - 2 is f^(-1)(x) = 9x + 18.

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

A: A function has an inverse if it is one-to-one, meaning that each output value corresponds to exactly one input value. In other words, if a function passes the horizontal line test, it has an inverse.

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

A: A function and its inverse are like two sides of the same coin. The function maps an input x to an output f(x), while the inverse function maps the output f(x) back to the input x.

Q: Can a function have more than one inverse?

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

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

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

  1. Swap the x and y variables: Swap the x and y variables in the original function.
  2. Solve for y: Solve for y in the equation x = (1/9)y^2 - 2y.
  3. Multiply both sides by 9: Multiply both sides of the equation by 9 to isolate y.
  4. Simplify the equation: Simplify the equation to get the inverse function.

Q: What is the inverse function of f(x) = x^2 + 2x?

A: The inverse function of f(x) = x^2 + 2x is f^(-1)(x) = (1/2)(x - 2) + 1.

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. The inverse function is only defined for one-to-one functions.

Q: How do I know if a function is one-to-one?

A: A function is one-to-one if it passes the horizontal line test. In other words, if a horizontal line intersects the graph of the function at most once, then the function is one-to-one.

Q: What is the significance of the inverse function?

A: The inverse function is significant because it allows us to solve equations and find the value of a function at a given point. It is also used in many real-world applications, such as physics, engineering, and economics.