For The Function \[$ K(x) = -16x - 8 \$\], What Is The Inverse Function \[$ K^{-1}(x) \$\]?A. \[$ K(x) = -16x - 8 \$\]B. \[$ K(x) = -16(x - 8) \$\]C. \[$ K(x) = -16x + 8 \$\]D. \[$ K^{-1}(x) = -16(x + 8) \$\]

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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) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. In this article, we will focus on finding the inverse function of a linear function, specifically the function k(x) = -16x - 8.

What is a Linear Function?

A linear function is a function that can be written in the form f(x) = mx + b, where m is the slope of the function and b is the y-intercept. The graph of a linear function is a straight line. In the case of the function k(x) = -16x - 8, the slope is -16 and the y-intercept is -8.

Finding the Inverse Function

To find the inverse function of k(x) = -16x - 8, we need to follow these steps:

  1. Swap the x and y variables: We start by swapping the x and y variables in the function k(x) = -16x - 8. This gives us x = -16y - 8.
  2. Solve for y: Next, we need to solve for y in the equation x = -16y - 8. To do this, we add 8 to both sides of the equation, which gives us x + 8 = -16y.
  3. Divide both sides by -16: Finally, we divide both sides of the equation x + 8 = -16y by -16, which gives us y = -\frac{x + 8}{16}.
  4. Write the inverse function: The inverse function of k(x) = -16x - 8 is k^(-1)(x) = -\frac{x + 8}{16}.

Simplifying the Inverse Function

We can simplify the inverse function k^(-1)(x) = -\frac{x + 8}{16} by multiplying both the numerator and denominator by -1. This gives us k^(-1)(x) = \frac{-x - 8}{16}.

Comparing the Options

Now that we have found the inverse function k^(-1)(x) = \frac{-x - 8}{16}, we can compare it to the options given in the problem.

  • Option A: k(x) = -16x - 8 is the original function, not the inverse function.
  • Option B: k(x) = -16(x - 8) is not the inverse function.
  • Option C: k(x) = -16x + 8 is not the inverse function.
  • Option D: k^(-1)(x) = -16(x + 8) is not the inverse function.

Conclusion

In conclusion, the inverse function of k(x) = -16x - 8 is k^(-1)(x) = \frac{-x - 8}{16}. This is the correct answer.

Final Answer

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) and its inverse f^(-1)(x), then f(f^(-1)(x)) = x and f^(-1)(f(x)) = x.

Q: Why do we need to find the inverse function?

A: We need to find the inverse function to solve equations that involve the original function. For example, if we have an equation of the form f(x) = y, we can use the inverse function to find the value of x.

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

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

  1. Swap the x and y variables in the function.
  2. Solve for y.
  3. Write the inverse function.

Q: What is the inverse function of k(x) = -16x - 8?

A: The inverse function of k(x) = -16x - 8 is k^(-1)(x) = \frac{-x - 8}{16}.

Q: How do we simplify the inverse function?

A: We can simplify the inverse function by multiplying both the numerator and denominator by -1.

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

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

  • Swapping the x and y variables incorrectly.
  • Solving for y incorrectly.
  • Writing the inverse function incorrectly.

Q: Can we find the inverse function of a quadratic function?

A: Yes, we can find the inverse function of a quadratic function. However, the process is more complex and may involve using the quadratic formula.

Q: Can we find the inverse function of a polynomial function?

A: Yes, we can find the inverse function of a polynomial function. However, the process may involve using algebraic techniques such as factoring and simplifying.

Q: Can we find the inverse function of a rational function?

A: Yes, we can find the inverse function of a rational function. However, the process may involve using algebraic techniques such as simplifying and canceling.

Q: What are some real-world applications of inverse functions?

A: Some real-world applications of inverse functions include:

  • Physics: Inverse functions are used to describe the motion of objects under the influence of forces.
  • Engineering: Inverse functions are used to design and optimize systems.
  • Economics: Inverse functions are used to model the behavior of economic systems.

Conclusion

In conclusion, inverse functions are an important concept in mathematics that have many real-world applications. By understanding how to find the inverse function of a linear function, we can solve equations and model real-world phenomena.

Final Answer

The final answer is: −x−816\boxed{\frac{-x - 8}{16}}