Math Problem Statement

Given the functions f(x) = 3x + 4 and g(x) = 5x^2 + 2x, what is the simplified expression for the composition (f o g)(x)?

Solution

In the image you uploaded, you have the functions f(x)=3x+4f(x) = 3x + 4 and g(x)=5x2+2xg(x) = 5x^2 + 2x. You are tasked with finding the composition of the functions, denoted as (fg)(x)(f \circ g)(x), which means f(g(x))f(g(x)).

Let's calculate f(g(x))f(g(x)):

  1. Substitute g(x)g(x) into f(x)f(x): f(g(x))=f(5x2+2x)f(g(x)) = f(5x^2 + 2x)

  2. Replace xx in f(x)f(x) with 5x2+2x5x^2 + 2x: f(5x2+2x)=3(5x2+2x)+4f(5x^2 + 2x) = 3(5x^2 + 2x) + 4

  3. Distribute and simplify: 3(5x2+2x)+4=15x2+6x+43(5x^2 + 2x) + 4 = 15x^2 + 6x + 4

So, the composition (fg)(x)(f \circ g)(x) is 15x2+6x+415x^2 + 6x + 4.

If you have any further questions or need more details on this process, feel free to ask!

Related Questions:

  1. What is the graphical interpretation of the composition of functions?
  2. Can you determine (gf)(x)(g \circ f)(x) using the same functions?
  3. What happens if the order of composition is reversed?
  4. How do compositions affect the domain and range of the resulting function?
  5. Is there a real-life scenario where composition of functions is applied?

Tip:

When solving for a function composition, always substitute the entire function for each instance of xx in the outer function and then simplify.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Polynomial Operations

Formulas

f(g(x)) = f(h(x)) for g(x) = h(x)

Theorems

Composition of functions

Suitable Grade Level

College