Suppose f(x) =1𝑥x1​ then (𝑓∘𝑓∘𝑓)(𝑥)(f∘f∘f)(x) =A.11𝑥x1​ 1​ B.1𝑥x1​ C.𝑥2x 2 D.1𝑥2x 2 1​ E.xSUBMITarrow_backPREVIOUS

1. Break Down the Problem

We start with the function $f(x) = \frac{1}{x}$. We need to find $(f \circ f \circ f)(x)$. This means applying the function $f$ three times in succession.

2. Relevant Concepts

The composition of functions, $(f \circ g)(x)$, is defined as $f(g(x))$. Specifically, we will evaluate the inner function and then keep applying $f$.

3. Analysis and Detail

1. Calculate $f(f(x))$: $f(f(x)) = f\left(\frac{1}{x}\right) = \frac{1}{\frac{1}{x}} = x$

2. Calculate $f(f(f(x)))$: $f(f(f(x))) = f(f(f(x))) = f(x) = \frac{1}{x}$

4. Verify and Summarize

After performing the function composition three times, we find: $(f \circ f \circ f)(x) = \frac{1}{x}$ Thus, $(f \circ f \circ f)(x) = f(x)$.

Final Answer

The final result is $(f \circ f \circ f)(x) = \frac{1}{x}$, which corresponds to option A. $\frac{1}{x}$.