f(x)=x-2/x+2 ;g(x)1/x$$ ,\left(f.g\right)\left(x\right)= $$

You want to find the composition of the functions $ f(x) = \frac{x - 2}{x + 2} $ and $ g(x) = \frac{1}{x} $, specifically $\left( f \circ g \right)(x)$. This notation represents $ f(g(x)) $.

To find $ \left( f \circ g \right)(x) $, substitute $ g(x) = \frac{1}{x} $ into the function $ f(x) $:

\[
(f \circ g)(x) = f\left(g(x)\right) = f\left(\frac{1}{x}\right)
\]

Given that $ f(x) = \frac{x - 2}{x + 2} $, substitute $ \frac{1}{x} $ for $ x $:

\[
f\left(\frac{1}{x}\right) = \frac{\frac{1}{x} - 2}{\frac{1}{x} + 2}
\]

Now, let's simplify this expression:

\[
f\left(\frac{1}{x}\right) = \frac{\frac{1 - 2x}{x}}{\frac{1 + 2x}{x}}
\]

Since the denominator $ x $ cancels out:

\[
f\left(\frac{1}{x}\right) = \frac{1 - 2x}{1 + 2x}
\]

Therefore, the composition $ \left( f \circ g \right)(x) $ is:

\[
\left( f \circ g \right)(x) = \frac{1 - 2x}{1 + 2x}
\]

Do you have any questions or would you like more details on this topic?

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Here are some related questions you might find helpful:

1. What is the domain of the composition $ \left( f \circ g \right)(x) $?
2. How do you find the inverse of $ \left( f \circ g \right)(x) $?
3. Can you determine the range of $ \left( f \circ g \right)(x) $?
4. What happens if you compute the composition in the reverse order, $ \left( g \circ f \right)(x) $?
5. How would you solve for $ x $ if $ \left( f \circ g \right)(x) = 0 $?

**Tip:** Always remember that when composing functions, you must consider the domains of both functions to ensure the composition is valid over the appropriate intervals.

Explore the composition of functions f(x) = (x - 2) / (x + 2) and g(x) = 1 / x. Learn how to find (f ∘ g)(x), step-by-step, with a detailed solution. Suitable for high school students studying functions and composition in algebra.

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