Express the function in the form

*f* ∘ *g*.

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

*G*(*x*) =

cube root (x/2+x)

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

### Step-by-step decomposition:

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

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

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

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

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 \( f \circ g \) 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.

Express the function in the form f ∘ g. (Use non-identity functions for f and g.) G(x) = cube root (x/2 + x)

Learn how to express the function G(x) = cube root(x/2 + x) in the form f ∘ g, using function composition. This problem explores key concepts like cube roots and algebraic manipulation. Suitable for students in Grades 10-12.

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