Math Problem Statement

Let f ( x )

2 x + 2 and g ( x )

3 x 2 + 3 x . After simplifying, ( f ∘ g ) ( x )

Solution

Given the functions:

f(x)=2x+2f(x) = 2x + 2 and g(x)=3x2+3xg(x) = 3x^2 + 3x,

we want to find (fg)(x)(f \circ g)(x), which means applying ff to the result of g(x)g(x). This can be written as:

(fg)(x)=f(g(x)).(f \circ g)(x) = f(g(x)).

Step 1: Substitute g(x)g(x) into f(x)f(x)

Since g(x)=3x2+3xg(x) = 3x^2 + 3x, we substitute this into f(x)f(x):

f(g(x))=f(3x2+3x).f(g(x)) = f(3x^2 + 3x).

Now, using the definition of f(x)f(x), which is f(x)=2x+2f(x) = 2x + 2, we replace xx in f(x)f(x) with 3x2+3x3x^2 + 3x:

f(3x2+3x)=2(3x2+3x)+2.f(3x^2 + 3x) = 2(3x^2 + 3x) + 2.

Step 2: Simplify the expression

Distribute the 2:

f(3x2+3x)=23x2+23x+2.f(3x^2 + 3x) = 2 \cdot 3x^2 + 2 \cdot 3x + 2.

This gives:

f(3x2+3x)=6x2+6x+2.f(3x^2 + 3x) = 6x^2 + 6x + 2.

Thus, the simplified form of (fg)(x)(f \circ g)(x) is:

(fg)(x)=6x2+6x+2.(f \circ g)(x) = 6x^2 + 6x + 2.


Do you need further clarification or details on any step?

Here are 5 follow-up questions to consider:

  1. How would the result change if we switch the composition to gfg \circ f?
  2. What happens if we differentiate (fg)(x)(f \circ g)(x)?
  3. Can we compose more functions in a chain, like h(x)f(x)g(x)h(x) \circ f(x) \circ g(x)?
  4. How would the composition (fg)(x)(f \circ g)(x) behave for negative values of xx?
  5. What are the general properties of function composition?

Tip: When composing functions, always substitute one function's output into the other carefully and simplify step by step.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Algebra
Polynomial Functions

Formulas

f(x) = 2x + 2
g(x) = 3x^2 + 3x
(f ∘ g)(x) = f(g(x))

Theorems

Function Composition Rule

Suitable Grade Level

Grades 9-12