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 , 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:
We need to complete the square for the quadratic term. To do that:
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Take half of the coefficient of , which is , and divide by 2:
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Square this value:
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Add and subtract 81 inside the expression to complete the square:
Thus, the function becomes:
Step 2: Solve for the inverse
Now, swap and to solve for the inverse function:
Add 81 to both sides:
Take the square root of both sides:
Finally, solve for :
Step 3: Interpretation and solving for b, c, d
To find the unknown values , , and , the problem should have provided values for that correspond to , , and . Can you clarify or provide the values of that relate to each? This way, we can substitute them into the inverse function to find , , and .
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:
- How can we verify that this inverse function is correct?
- How do we determine the domain and range of the inverse function?
- What is the significance of completing the square in this process?
- How does the inverse of a quadratic differ from linear functions?
- 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