$$ f(x)=\frac{x}{x+2} $$ then find $$ f^{2} $$

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UpStudy AI · Accepted Answer

To find $$ f^{2}(x) $$ for the function $$ f(x) = \frac{x}{x + 2} $$, we need to clarify the notation:

- **$$ f^{2}(x) $$** typically denotes the composition of the function $$ f $$ with itself, i.e., $$ f(f(x)) $$.
- Alternatively, it can sometimes represent the square of the function, i.e., $$ \left( f(x) \right)^2 $$.

Below are both interpretations:

### 1. Composition $$ f(f(x)) $$

Let's compute $$ f(f(x)) $$:

$$
f(f(x)) = f\left( \frac{x}{x + 2} \right) = \frac{\frac{x}{x + 2}}{\frac{x}{x + 2} + 2}
$$

Simplify the denominator:

$$
\frac{x}{x + 2} + 2 = \frac{x + 2(x + 2)}{x + 2} = \frac{3x + 4}{x + 2}
$$

Now, substitute back:

$$
f(f(x)) = \frac{\frac{x}{x + 2}}{\frac{3x + 4}{x + 2}} = \frac{x}{3x + 4}
$$

So,

$$
f^{2}(x) = \frac{x}{3x + 4}
$$

### 2. Square of the Function $$ \left( f(x) \right)^2 $$

If $$ f^{2}(x) $$ represents the square of $$ f(x) $$, then:

$$
\left( f(x) \right)^2 = \left( \frac{x}{x + 2} \right)^2 = \frac{x^2}{(x + 2)^2}
$$

### Conclusion

Depending on the intended meaning:

- **If $$ f^{2}(x) $$ means composition:** 
  $$
  f^{2}(x) = \frac{x}{3x + 4}
  $$
  
- **If $$ f^{2}(x) $$ means squaring the function:** 
  $$
  f^{2}(x) = \frac{x^2}{(x + 2)^2}
  $$

Given the common interpretation in function notation, it's likely that **$$ f^{2}(x) = \frac{x}{3x + 4} $$** is the expected answer.

**Final Answer:**
After simplifying, the second iterate is x divided by (3 x plus four). Thus,

f²(x) = x ⁄ (3x + 4)
$$ f^{2}(x) = \frac{x}{3x + 4} $$

\( f(x)=\frac{x}{x+2} \) then find \( f^{2} \)

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