Math Problem Statement

Given g(f(x)) = 4x^2 + 8x + 1 and f(x) = 2x + 1, determine g(x).

Solution

To solve the given problem, let's break it down:

Problem Interpretation

We are given:

  • g(f(x))=4x2+8x+1g(f(x)) = 4x^2 + 8x + 1
  • f(x)=2x+1f(x) = 2x + 1

We need to find g(x)g(x).

Solution

  1. Since g(f(x))=4x2+8x+1g(f(x)) = 4x^2 + 8x + 1 and f(x)=2x+1f(x) = 2x + 1, substitute f(x)f(x) with 2x+12x + 1 in the expression for gg: g(2x+1)=4x2+8x+1g(2x + 1) = 4x^2 + 8x + 1

  2. Let u=2x+1u = 2x + 1, so we need to express g(u)g(u) in terms of uu.

  3. Solve for xx in terms of uu: u=2x+1x=u12u = 2x + 1 \Rightarrow x = \frac{u - 1}{2}

  4. Substitute x=u12x = \frac{u - 1}{2} into 4x2+8x+14x^2 + 8x + 1: g(u)=4(u12)2+8(u12)+1g(u) = 4\left(\frac{u - 1}{2}\right)^2 + 8\left(\frac{u - 1}{2}\right) + 1

  5. Simplify this expression:

    • Compute (u12)2=(u1)24=u22u+14\left(\frac{u - 1}{2}\right)^2 = \frac{(u - 1)^2}{4} = \frac{u^2 - 2u + 1}{4}
    • So, 4(u12)2=u22u+14\left(\frac{u - 1}{2}\right)^2 = u^2 - 2u + 1
    • And, 8(u12)=4(u1)=4u48\left(\frac{u - 1}{2}\right) = 4(u - 1) = 4u - 4
    • Therefore

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Composition
Quadratic Functions

Formulas

g(f(x)) = g(2x + 1) = 4x^2 + 8x + 1
u = 2x + 1
x = (u - 1)/2

Theorems

Function Composition

Suitable Grade Level

Grades 9-11