To determine whether [tex]\( f \circ f(x) \)[/tex] is an identity function when [tex]\( f(x) = \frac{1}{1-x} \)[/tex], we need to find [tex]\( f(f(x)) \)[/tex] and check if it equals [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Define the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{1}{1-x}
\][/tex]
2. Calculate [tex]\( f(f(x)) \)[/tex]:
To find [tex]\( f(f(x)) \)[/tex], we need to substitute [tex]\( f(x) \)[/tex] into itself. First, find [tex]\( f(f(x)) \)[/tex] by substituting [tex]\( \frac{1}{1-x} \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( f \)[/tex].
[tex]\[
f(f(x)) = f\left(\frac{1}{1-x}\right)
\][/tex]
3. Simplify the expression:
We substitute [tex]\( \frac{1}{1-x} \)[/tex] into the original function [tex]\( f(x) \)[/tex].
[tex]\[
f\left(\frac{1}{1-x}\right) = \frac{1}{1 - \left(\frac{1}{1-x}\right)}
\][/tex]
4. Simplify the inner expression:
We need to simplify the denominator.
[tex]\[
1 - \frac{1}{1-x} = \frac{(1-x) - 1}{1-x} = \frac{1-x - 1}{1-x} = \frac{-x}{1-x}
\][/tex]
5. Complete the substitution:
Now that we have the simplified denominator, substitute it back into the equation.
[tex]\[
f\left(\frac{1}{1-x}\right) = \frac{1}{\frac{-x}{1-x}} = \frac{1-x}{-x} = -\frac{1-x}{x} = \frac{x-1}{x}
\][/tex]
Therefore, we have:
[tex]\[
f(f(x)) = \frac{x-1}{x}
\][/tex]
6. Check if [tex]\( f(f(x)) = x \)[/tex]:
For [tex]\( f \circ f(x) \)[/tex] to be an identity function, [tex]\( f(f(x)) \)[/tex] must equal [tex]\( x \)[/tex].
[tex]\[
\frac{x-1}{x} \neq x
\][/tex]
Clearly, [tex]\( \frac{x-1}{x} \)[/tex] is not equal to [tex]\( x \)[/tex].
### Conclusion:
[tex]\[
f(f(x)) = \frac{x-1}{x} \neq x
\][/tex]
Thus, [tex]\( f \circ f(x) \)[/tex] is not an identity function.
