If `f:R rarr R` defined by `f(x)=(x-1)/(2)`, find `(f o f)(x)`

Answer

To find \( (f \circ f)(x) \) for the function \( f(x) = \frac{x - 1}{2} \), we will follow these steps:

### Step 1: Understand the composition of functions
The composition of functions \( (f \circ f)(x) \) means we need to apply the function \( f \) to the result of \( f(x) \). This can be expressed as \( f(f(x)) \).

### Step 2: Calculate \( f(x) \)
Given the function:
\[ f(x) = \frac{x - 1}{2} \]

### Step 3: Substitute \( f(x) \) into itself
Now we need to find \( f(f(x)) \):
\[ f(f(x)) = f\left(\frac{x - 1}{2}\right) \]

### Step 4: Replace \( x \) in \( f(x) \) with \( \frac{x - 1}{2} \)
Using the definition of \( f(x) \):
\[ f\left(\frac{x - 1}{2}\right) = \frac{\left(\frac{x - 1}{2}\right) - 1}{2} \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ = \frac{\frac{x - 1}{2} - 1}{2} \]
To combine the terms in the numerator:
\[ = \frac{\frac{x - 1}{2} - \frac{2}{2}}{2} = \frac{\frac{x - 1 - 2}{2}}{2} = \frac{x - 3}{2 \cdot 2} = \frac{x - 3}{4} \]

### Step 6: Write the final result
Thus, we have:
\[ (f \circ f)(x) = \frac{x - 3}{4} \]

### Final Answer
\[ (f \circ f)(x) = \frac{x - 3}{4} \]

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