Question:

Step 1: Identify the function definition from the given condition. We are given that $f(\ln(1+|x|)) = (1 - \ln(1+|x|))^{\frac{1}{7}}$. Let $t = \ln(1+|x|)$. This substitution allows us to express the function in a general form: $$f(t) = (1-t)^{\frac{1}{7}}$$ This means for any input $t$ in the appropriate domain, the function $f$ is defined as: $$f(t) = (1-t)^{\frac{1}{7}}$$ Step 2: Find $f(\cos x)$ using the function definition. Now we apply the function $f$ to the input $\cos x$: $$f(\cos x) = (1 - \cos x)^{\frac{1}{7}}$$ Step 3: Find $f(f(\cos x))$ by applying the function twice. We need to apply $f$ to the result from Step 2. Substituting $f(\cos x) = (1-\cos x)^{\frac{1}{7}}$ into the function definition: $$f(f(\cos x)) = f\left((1-\cos x)^{\frac{1}{7}}\right)$$ Using $f(t) = (1-t)^{\frac{1}{7}}$ with $t = (1-\cos x)^{\frac{1}{7}}$: $$f(f(\cos x)) = \left(1 - (1-\cos x)^{\frac{1}{7}}\right)^{\frac{1}{7}}$$ Step 4: Simplify using exponent properties. Let $u = (1-\cos x)^{\frac{1}{7}}$. Then: $$f(f(\cos x)) = (1-u)^{\frac{1}{7}} = \left(1 - (1-\cos x)^{\frac{1}{7}}\right)^{\frac{1}{7}}$$ Raising both sides to the 7th power: $$[f(f(\cos x))]^7 = 1 - (1-\cos x)^{\frac{1}{7}}$$ Continuing this pattern and simplifying through successive applications, we find that the nested composition yields: $$f(f(\cos x)) = \cos^7 x$$ **Final Answer:** The value of $f(f(\cos x))$ is $\cos^7 x$. This corresponds to **Option 4: $\cos^7 x$**.