Math Problem Statement

solve

A quadratic function is given below: f(x) = 2(1/2(x - 3))^2 + 4 If the domain of f(x) is restricted to x ≤ 3, which function best represents f^(-1)(x)? Options: A: f^(-1)(x) = sqrt(x - 4) + 3 B: f^(-1)(x) = -sqrt(1/2(x - 4)) + 3 C: f^(-1)(x) = -2sqrt(1/2(x - 4)) + 3 D: f^(-1)(x) = sqrt(x - 4) + 3

Solution

Problem Analysis

We are given a quadratic function:
f(x)=2(12(x3))2+4f(x) = 2\left(\frac{1}{2}(x - 3)\right)^2 + 4
and are tasked to find the inverse function f1(x)f^{-1}(x) when the domain of f(x)f(x) is restricted to x3x \leq 3.

The inverse function reverses the input-output relationship of f(x)f(x), meaning if y=f(x)y = f(x), then x=f1(y)x = f^{-1}(y).


Step 1: Simplify the given quadratic function

The function is:
f(x)=2(12(x3))2+4f(x) = 2\left(\frac{1}{2}(x - 3)\right)^2 + 4

Simplify the term inside the parentheses:
f(x)=2(x32)2+4f(x) = 2\left(\frac{x - 3}{2}\right)^2 + 4

f(x)=2((x3)24)+4f(x) = 2\left(\frac{(x - 3)^2}{4}\right) + 4

f(x)=(x3)22+4f(x) = \frac{(x - 3)^2}{2} + 4


Step 2: Solve for the inverse

Let f(x)=yf(x) = y. Replace f(x)f(x) with yy:
y=(x3)22+4y = \frac{(x - 3)^2}{2} + 4

Rearrange to isolate (x3)2(x - 3)^2:
y4=(x3)22y - 4 = \frac{(x - 3)^2}{2}

2(y4)=(x3)22(y - 4) = (x - 3)^2

Take the square root of both sides, considering the domain restriction x3x \leq 3:
x3=2(y4)x - 3 = -\sqrt{2(y - 4)}

x=2(y4)+3x = -\sqrt{2(y - 4)} + 3

Thus, the inverse function is:
f1(x)=2(x4)+3f^{-1}(x) = -\sqrt{2(x - 4)} + 3


Step 3: Match the correct option

The correct answer is:
B: f1(x)=12(x4)+3\boxed{\text{B: } f^{-1}(x) = -\sqrt{\frac{1}{2}(x - 4)} + 3}


Would you like further details or explanations?


Related Questions:

  1. How do you verify if a function and its inverse are correct?
  2. What happens to the domain and range when finding an inverse of a function?
  3. How do domain restrictions affect the validity of an inverse function?
  4. Why do we only consider the negative root for x3x \leq 3?
  5. 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