If
ℎ
(
𝑥
)
=
(
1
𝑥
+
2
)
(
1
𝑥
+
3
)
, find functions
𝑓
and
𝑔
such that
ℎ
(
𝑥
)
=
(
𝑓
∘
𝑔
)
(
𝑥
)

Given the function \( h(x) = \left( \frac{1}{x} + 2 \right) \left( \frac{1}{x} + 3 \right) \), we want to find functions \( f \) and \( g \) such that \( h(x) = (f \circ g)(x) \).

To solve this, let's assume \( g(x) \) is a simple function that transforms \( x \) into something that both \( f \) and \( h \) can work with easily.

Let's start with:
\[ g(x) = \frac{1}{x} \]

Then, \( g(x) = \frac{1}{x} \) transforms \( x \) into \( \frac{1}{x} \).

Next, we need to find a function \( f \) such that:
\[ h(x) = f(g(x)) = f\left(\frac{1}{x}\right) \]

Given \( h(x) = \left( \frac{1}{x} + 2 \right) \left( \frac{1}{x} + 3 \right) \), we can see that \( f \) should take \( \frac{1}{x} \) and turn it into \( \left( \frac{1}{x} + 2 \right) \left( \frac{1}{x} + 3 \right) \).

Thus, we define \( f(u) \) as:
\[ f(u) = (u + 2)(u + 3) \]

Now, substituting \( u = \frac{1}{x} \) into \( f(u) \):
\[ f\left(\frac{1}{x}\right) = \left( \frac{1}{x} + 2 \right) \left( \frac{1}{x} + 3 \right) \]

Therefore, the functions \( f \) and \( g \) are:
\[ f(u) = (u + 2)(u + 3) \]
\[ g(x) = \frac{1}{x} \]

So,
\[ h(x) = (f \circ g)(x) = f\left(g(x)\right) = f\left(\frac{1}{x}\right) = \left( \frac{1}{x} + 2 \right) \left( \frac{1}{x} + 3 \right) \]

This completes the solution.

Do you want details or have any questions?

Here are 5 numbered questions you can ask:
1. How do you compose functions?
2. Can you explain the composition of functions with another example?
3. What are some properties of function composition?
4. How do you find the inverse of a function?
5. Can you explain how to simplify composed functions?

Tip: When finding functions \( f \) and \( g \) for composition, it's helpful to start with simple transformations and gradually build up to match the desired function.

Explore how to find functions f and g such that h(x) = (f ∘ g)(x), given h(x) = ((1/x) + 2)((1/x) + 3). Learn step-by-step with detailed algebraic expressions and transformations. Suitable for high school students.

Math Problem Statement

If ℎ ( 𝑥 )

( 1 𝑥 + 2 ) ( 1 𝑥 + 3 ) , find functions 𝑓 and 𝑔 such that ℎ ( 𝑥 )

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