Find functions f(x) and g(x) so the given function can be expressed as 
h(x) = f(g(x)).
 (Use non-identity functions for 
f(x) and g(x).)
h(x) = 
2
x − 6
{f(x), g(x)}=

To express the function \( h(x) = 2x - 6 \) as \( h(x) = f(g(x)) \) using non-identity functions \( f(x) \) and \( g(x) \), we need to find two functions \( f(x) \) and \( g(x) \) such that:

\[
h(x) = f(g(x))
\]

A common strategy is to let \( g(x) \) be a simpler function related to the argument of \( h(x) \), and then find \( f(x) \) accordingly. Here is one possible solution:

Let:
\[
g(x) = x - 3
\]

Then:
\[
f(x) = 2x
\]

Now, let's verify:

\[
h(x) = f(g(x)) = f(x - 3) = 2(x - 3) = 2x - 6
\]

This matches the original function \( h(x) = 2x - 6 \).

Thus, the functions \( f(x) \) and \( g(x) \) are:

\[
f(x) = 2x, \quad g(x) = x - 3
\]

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How can the composition of functions be verified?
2. What other pairs of functions could be used to express \( h(x) = 2x - 6 \) as \( f(g(x)) \)?
3. How would you express a quadratic function as a composition of two functions?
4. What is the importance of using non-identity functions in composition?
5. Can the same approach be used to decompose trigonometric functions?

**Tip:** When decomposing functions into compositions, start by simplifying or isolating parts of the function that can be treated as \( g(x) \), and then find the corresponding \( f(x) \).

Find functions f(x) and g(x) so the given function can be expressed as h(x) = f(g(x)). (Use non-identity functions for f(x) and g(x).) h(x) = 2x - 6 {f(x), g(x)}=

Learn how to express the linear function h(x) = 2x - 6 as a composition of two functions f(x) and g(x). This solution demonstrates the use of function composition in algebra, with non-identity functions. Suitable for students in Grades 9-11.

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