Find functions f(x) and g(x) so that h(x)=√(5x+10) can be expressed as h(x)=f(g(x))

Pick f and g such that f(x)≠x and g(x)≠x

Enter your functions as a list [f(x),g(x)]

To express \( h(x) = \sqrt{5x + 10} \) as \( h(x) = f(g(x)) \), we need to decompose the function into two parts, where:

- \( g(x) \) is an inner function that simplifies the argument of the square root.
- \( f(x) \) is an outer function that applies the square root to the result of \( g(x) \).

### Decomposition:
We can choose \( g(x) = 5x + 10 \) and \( f(x) = \sqrt{x} \). This gives:

\[
h(x) = f(g(x)) = \sqrt{5x + 10}
\]

Thus, the functions are:
\[
f(x) = \sqrt{x}, \quad g(x) = 5x + 10
\]

### Final Answer:
\[
[f(x), g(x)] = \left[ \sqrt{x}, 5x + 10 \right]
\]

Let me know if you'd like more details on this decomposition or have any questions!

#### Related Questions:
1. How would you express \( h(x) = (3x + 7)^2 \) as \( f(g(x)) \)?
2. Can you decompose \( h(x) = \ln(2x + 1) \) into functions \( f \) and \( g \)?
3. What functions \( f \) and \( g \) would work for \( h(x) = \sin(4x - 3) \)?
4. How can you express \( h(x) = e^{6x+2} \) as \( f(g(x)) \)?
5. What happens if you try to decompose \( h(x) = x^2 + 5x + 4 \) into \( f(g(x)) \)?

### Tip:
When decomposing functions, think about simplifying the most complex operations first, such as square roots, exponentials, or logarithms.

Find functions f(x) and g(x) so that h(x)=√(5x+10) can be expressed as h(x)=f(g(x)). Pick f and g such that f(x)≠x and g(x)≠x.

Learn how to express the function h(x)=√(5x+10) as a composition of two functions f(x) and g(x). This solution breaks down the function using concepts from algebra and function composition, providing a detailed explanation for high school students in Grades 10-12.

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