Math Problem Statement
Let the function f : [a, ∞) → [1, ∞) be given by f (x) = x2 − 4x + 5. (a) Show that f is a surjection for a = 0. (b) Determine the smallest value of a such that f is injective. (c) Determine for the value of a that you calculated in (b) an expression for the inverse function f −1.
Solution
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Surjection
Injection
Inverse Functions
Formulas
f(x) = x^2 - 4x + 5
f(x) = (x - 2)^2 + 1
f'(x) = 2(x - 2)
f^{-1}(y) = 2 + √(y - 1)
Theorems
Surjection and Injection Theorems
Completing the Square
Suitable Grade Level
Grades 11-12
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