Math Problem Statement

Jika f(x) = x + 2 dan g(x) = 3x^2 + 4x + 1, maka (gof)(x) adalah....

Solution

The question asks to find (gf)(x)(g \circ f)(x), which means g(f(x))g(f(x)). Given the functions:

  • f(x)=x+2f(x) = x + 2
  • g(x)=3x2+4x+1g(x) = 3x^2 + 4x + 1

We need to compute g(f(x))g(f(x)), which involves substituting f(x)f(x) into g(x)g(x).

Step-by-step Solution:

  1. Start by substituting f(x)=x+2f(x) = x + 2 into g(x)g(x): g(f(x))=g(x+2)g(f(x)) = g(x + 2)

  2. Now, substitute x+2x + 2 wherever there is xx in g(x)=3x2+4x+1g(x) = 3x^2 + 4x + 1: g(x+2)=3(x+2)2+4(x+2)+1g(x + 2) = 3(x + 2)^2 + 4(x + 2) + 1

  3. Expand (x+2)2(x + 2)^2: (x+2)2=x2+4x+4(x + 2)^2 = x^2 + 4x + 4 So, g(x+2)=3(x2+4x+4)+4(x+2)+1g(x + 2) = 3(x^2 + 4x + 4) + 4(x + 2) + 1

  4. Expand the terms: g(x+2)=3x2+12x+12+4x+8+1g(x + 2) = 3x^2 + 12x + 12 + 4x + 8 + 1

  5. Combine like terms: g(x+2)=3x2+(12x+4x)+(12+8+1)g(x + 2) = 3x^2 + (12x + 4x) + (12 + 8 + 1) g(x+2)=3x2+16x+21g(x + 2) = 3x^2 + 16x + 21

Thus, (gf)(x)=3x2+16x+21(g \circ f)(x) = 3x^2 + 16x + 21.

The correct answer is option E: 3x2+16x+213x^2 + 16x + 21.

Would you like more details on any part of this process? Here are five related questions to deepen your understanding:

  1. How is the composition of functions defined?
  2. What is the general method to simplify (x+a)2(x + a)^2?
  3. How can the distributive property help in expanding expressions like 3(x+2)23(x + 2)^2?
  4. Can the process of function composition be reversed? How would fg(x)f \circ g(x) differ here?
  5. How would you approach the same problem if g(x)g(x) had a cubic term?

Tip: When composing functions, always make sure to substitute the entire expression for f(x)f(x) correctly into g(x)g(x).

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomials
Algebra

Formulas

g(f(x)) = g(x+2)
(x+a)^2 = x^2 + 2ax + a^2

Theorems

Function Composition Theorem
Distributive Property

Suitable Grade Level

Grades 10-12