Math Problem Statement

Jika f(x) = 2x - 1 dan (f ∘ g)(x) = -14x + 21, nilai dari g(-3) = ....

Solution

The problem reads:

Jika f(x)=2x1f(x) = 2x - 1 dan (fg)(x)=14x+21(f \circ g)(x) = -14x + 21, nilai dari g(3)g(-3) = ....

Translation:

If f(x)=2x1f(x) = 2x - 1 and (fg)(x)=14x+21(f \circ g)(x) = -14x + 21, find g(3)g(-3).

Solution:

  1. (fg)(x)(f \circ g)(x) means f(g(x))f(g(x)).

    • This implies that f(g(x))=14x+21f(g(x)) = -14x + 21.
  2. We know that f(x)=2x1f(x) = 2x - 1, so substitute g(x)g(x) into f(x)f(x): f(g(x))=2g(x)1f(g(x)) = 2g(x) - 1 Therefore: 2g(x)1=14x+212g(x) - 1 = -14x + 21

  3. Solve for g(x)g(x): 2g(x)=14x+21+12g(x) = -14x + 21 + 1 2g(x)=14x+222g(x) = -14x + 22 g(x)=14x+222g(x) = \frac{-14x + 22}{2} g(x)=7x+11g(x) = -7x + 11

  4. Now, find g(3)g(-3): g(3)=7(3)+11g(-3) = -7(-3) + 11 g(3)=21+11=32g(-3) = 21 + 11 = 32

Thus, g(3)=32g(-3) = 32.

Would you like more details or further questions on this? Here are some related questions:

  1. What is the general process to find the composition of two functions?
  2. How do you check if your solution for g(x)g(x) is correct?
  3. How would the process change if f(x)f(x) was a quadratic function?
  4. Can (fg)(x)(f \circ g)(x) be defined for all functions ff and gg?
  5. What is the importance of inverse functions when dealing with compositions?

Tip: When solving function composition problems, always make sure to carefully substitute the expressions and simplify systematically to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Linear Equations

Formulas

f(x) = 2x - 1
(f ∘ g)(x) = -14x + 21
g(x) = -7x + 11

Theorems

Function Composition

Suitable Grade Level

Grades 10-12