Math Problem Statement

Express the function in the form

f ∘ g.

(Use non-identity functions for f and g.)

G(x) =

cube root (x/2+x)

Solution

We are given the function G(x)=x2+x3G(x) = \sqrt[3]{\frac{x}{2} + x}, and we want to express it in the form fgf \circ g, which means we are looking for two functions f(x)f(x) and g(x)g(x) such that G(x)=f(g(x))G(x) = f(g(x)).

Step-by-step decomposition:

  1. Inner function g(x)g(x): This will represent the expression inside the cube root. g(x)=x2+x=3x2g(x) = \frac{x}{2} + x = \frac{3x}{2}

  2. Outer function f(x)f(x): This will apply the cube root operation to the result of g(x)g(x). f(x)=x3f(x) = \sqrt[3]{x}

Thus, we can express G(x)G(x) as: G(x)=f(g(x))=f(3x2)=3x23G(x) = f(g(x)) = f\left( \frac{3x}{2} \right) = \sqrt[3]{\frac{3x}{2}}

So the functions ff and gg are: f(x)=x3,g(x)=3x2f(x) = \sqrt[3]{x}, \quad g(x) = \frac{3x}{2}

Would you like further details or clarifications on this? Here are some related questions to explore:

  1. How do we identify the inner and outer functions for composition?
  2. What are the general properties of composition of functions?
  3. How does the cube root affect the domain of the function?
  4. How can we verify if the decomposition fgf \circ g is correct?
  5. Can we generalize this method to other root functions?

Tip: When composing functions, start by identifying the most complex operation (usually the outer function) and then work inward.

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Math Problem Analysis

Mathematical Concepts

Function Composition
Cube Root
Algebraic Manipulation

Formulas

G(x) = f(g(x))
g(x) = x/2 + x = 3x/2
f(x) = cube root(x)

Theorems

Function Composition

Suitable Grade Level

Grades 10-12