Express  each  function  as  the  composition  of  functions  f (g(x)) for (f-g)(2)

To express a function as the composition of two functions \( f(g(x)) \), we are typically asked to break down a given function into two separate functions, \( f \) and \( g \), such that when composed, they reproduce the original function.

The notation \( (f \circ g)(2) \), or \( f(g(2)) \), means that we first evaluate \( g(2) \), and then apply the function \( f \) to the result.

To proceed:

1. If you have a specific function in mind that needs to be expressed as \( f(g(x)) \), please share it. Otherwise, I will explain the general idea with an example.

Let's consider the following function for illustration:

\[ h(x) = (3x + 1)^2 \]

We can express \( h(x) \) as the composition \( f(g(x)) \) by defining:

- \( g(x) = 3x + 1 \)
- \( f(x) = x^2 \)

So, the composition would be:

\[
f(g(x)) = (g(x))^2 = (3x + 1)^2
\]

Now, to evaluate \( (f \circ g)(2) \), we first find \( g(2) \):

\[
g(2) = 3(2) + 1 = 7
\]

Then, apply \( f \) to this result:

\[
f(7) = 7^2 = 49
\]

Thus, \( (f \circ g)(2) = 49 \).

If you had a specific function in mind, let me know, and I can adjust the explanation to match it!

Would you like more details or have any questions? Here are five related questions:

1. Can you provide a specific function you want to decompose into \( f(g(x)) \)?
2. How would you decompose a more complex function, such as \( h(x) = \sin(x^2 + 1) \)?
3. What happens when you reverse the order of the composition, i.e., \( g(f(x)) \)?
4. Can all functions be written as the composition of two functions?
5. How does function composition relate to inverse functions?

**Tip**: When decomposing a function, try to identify the "inner" and "outer" operations, which can help in defining \( g(x) \) and \( f(x) \) clearly.

Express each function as the composition of functions f (g(x)) for (f-g)(2)

Learn how to express a function as a composition of two functions, f(g(x)), with a detailed example involving (f ∘ g)(2). Understand key algebraic concepts of function composition, suitable for students in Grades 10-12.

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