Math Problem Statement
Solution
We are tasked with finding the inverse of the function and matching it to the given form .
Step 1: Completing the square
To express the given quadratic in a form that can be inverted, we start by completing the square.
The original function is:
Complete the square: Thus, the equation becomes:
Step 2: Inverting the function
Now, solve for to find the inverse: Take the square root of both sides: Solve for :
Thus, the inverse function is:
Step 3: Compare with the form given
The inverse is now in the form . Comparing it to the standard form , we can match the coefficients:
Thus, the values are:
Would you like further clarification on any step? Here are some related questions to think about:
- How does completing the square help in solving quadratic equations?
- Why do we take the square root on both sides when inverting the function?
- What is the general procedure for finding the inverse of a quadratic function?
- How do you determine the positive or negative branch of the square root when inverting?
- Can you find the domain and range of the inverse function?
Tip: When completing the square, always remember to balance the equation by adding and subtracting the same value to maintain equality.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Quadratic Equations
Completing the Square
Formulas
Inverse of a quadratic function: Solve y = ax^2 + bx + c for x.
Completing the square: (x - h)^2 = x^2 - 2hx + h^2
Theorems
Completing the Square Theorem
Inverse Function Theorem
Suitable Grade Level
Grades 10-12
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