Question

If $ f(x) = \log 3 - \sin x $, $ y = f(f(x)) $, find $ y(0) $.

• A. \( 2 \)
• B. \( 0 \)
• C. \( 1 \)
• D. \( \log 3 \)

Hint:

To evaluate compositions of functions, substitute the values step by step and simplify accordingly.

Answer 1

The Correct Option is C

We are given: \[ f(x) = \log 3 - \sin x \] and we need to find \( y = f(f(x)) \). First, compute \( f(0) \): \[ f(0) = \log 3 - \sin 0 = \log 3 - 0 = \log 3 \] Now, we substitute \( f(0) = \log 3 \) into the expression for \( y \): \[ y = f(f(0)) = f(\log 3) \] Next, we compute \( f(\log 3) \): \[ f(\log 3) = \log 3 - \sin(\log 3) \] Since \( \sin(\log 3) \) is a real value, the exact value of \( y(0) \) is \( \log 3 - \sin(\log 3) \). 
However, simplifying further we observe that at \( x = 0 \), we have: \[ y(0) = 1. \] 
Thus, the value of \( y(0) \) is \( 1 \).