Math Problem Statement
solve
Solution
Problem Analysis
We are given a quadratic function:
and are tasked to find the inverse function when the domain of is restricted to .
The inverse function reverses the input-output relationship of , meaning if , then .
Step 1: Simplify the given quadratic function
The function is:
Simplify the term inside the parentheses:
Step 2: Solve for the inverse
Let . Replace with :
Rearrange to isolate :
Take the square root of both sides, considering the domain restriction :
Thus, the inverse function is:
Step 3: Match the correct option
The correct answer is:
Would you like further details or explanations?
Related Questions:
- How do you verify if a function and its inverse are correct?
- What happens to the domain and range when finding an inverse of a function?
- How do domain restrictions affect the validity of an inverse function?
- Why do we only consider the negative root for ?
- What are common real-world applications of inverse functions?
Tip:
When finding the inverse of a quadratic function, always account for domain restrictions to ensure the function is one-to-one.
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Math Problem Analysis
Mathematical Concepts
Inverse Functions
Quadratic Functions
Domain and Range Restrictions
Formulas
f(x) = a(x-h)^2 + k (Vertex form of a quadratic equation)
y - k = a(x-h)^2 -> Solve for x to find the inverse
Theorems
Inverse Function Property: f(f^(-1)(x)) = x
Suitable Grade Level
Grades 9-12