Math Problem Statement

Use the inverse of the function y = x2 − 18x to find the unknown values. b= c= d=

Solution

To find the inverse of the function y=x218xy = x^2 - 18x, we will first complete the square to express the function in a form that allows us to find its inverse. Here’s the step-by-step process:

Step 1: Express the function and complete the square

The given function is:

y=x218xy = x^2 - 18x

We need to complete the square for the quadratic term. To do that:

  1. Take half of the coefficient of xx, which is 18-18, and divide by 2: 182=9\frac{-18}{2} = -9

  2. Square this value: (9)2=81(-9)^2 = 81

  3. Add and subtract 81 inside the expression to complete the square: y=(x218x+81)81=(x9)281y = (x^2 - 18x + 81) - 81 = (x - 9)^2 - 81

Thus, the function becomes: y=(x9)281y = (x - 9)^2 - 81

Step 2: Solve for the inverse

Now, swap xx and yy to solve for the inverse function:

x=(y9)281x = (y - 9)^2 - 81

Add 81 to both sides:

x+81=(y9)2x + 81 = (y - 9)^2

Take the square root of both sides:

x+81=y9\sqrt{x + 81} = y - 9

Finally, solve for yy:

y=x+81+9y = \sqrt{x + 81} + 9

Step 3: Interpretation and solving for b, c, d

To find the unknown values bb, cc, and dd, the problem should have provided values for xx that correspond to y=by = b, y=cy = c, and y=dy = d. Can you clarify or provide the values of xx that relate to each? This way, we can substitute them into the inverse function to find bb, cc, and dd.


Do you have any questions or would you like further details on any part of the process?

Here are 5 related questions for deeper understanding:

  1. How can we verify that this inverse function is correct?
  2. How do we determine the domain and range of the inverse function?
  3. What is the significance of completing the square in this process?
  4. How does the inverse of a quadratic differ from linear functions?
  5. Can we always find an inverse for quadratic functions?

Tip: When solving for inverses of functions, it’s important to ensure that the function is one-to-one over the interval you're considering.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inverse Functions
Completing the Square
Quadratic Functions

Formulas

y = x^2 - 18x
(x - 9)^2 - 81
y = √(x + 81) + 9

Theorems

Inverse Function Theorem
Quadratic Equations

Suitable Grade Level

Grades 9-12