Understanding Function Composition with f ° f

To find the formula of f ° f, we need to compose a function with itself. Given a function defined by f(x) 4×x2/(x - 5), where x ≠ 5.

Step-by-Step Simplification

The function composition f°f is defined by substituting f(x) into itself:

f°f(x) f(f(x)) 4×4×x2/(x - 5)2/(4×x2/(x - 5) - 5)

Step 1: Simplifying the Composition Expression

A more simplified view of the composition is:

f°f(x) 16×x8/(x - 5)2/(4×x2/(x - 5) - 5)

Step 2: Further Simplification

Simplifying the numerator and denominator separately:

f°f(x) 18×x - 2/(x - 5)/(27-x)/(x - 5)

Then, combining the terms:

f°f(x) 18×x - 2/(27 - x)

The final simplified form is:

f°f(x) 18×x - 2/(27 - x)

Key Points and Considerations

1. **Disambiguation**: It’s important to clearly define your function in this format. The correct form for clarity should be f(x) 4×x2/(x - 5).

2. **Domain Constraints**: When working with compositions, always check the domain to ensure that the new function is well-defined. Here, the domain excludes x5 and x27.

3. **Using Parentheses for Clarity**: Always use parentheses to ensure that operations are properly grouped and understood. This is especially important in compositions involving quotients.