Question

Let $f(x) = \sin^{-1}x$ and $g(x) = x - 2$. To define the composite function $f \circ g$, the largest domain of $g(x)$ has to be}

• A. $[2, 5]$
• B. $[1, 3]$
• C. $[0, 2]$
• D. $[-1, 3]$
• E. $[-3, 3]$

Hint:

For inverse trigonometric functions like \( \sin^{-1} \) or \( \cos^{-1} \), the argument must always lie between $-1$ and $1$. Just set the inner function within these bounds to find the restricted domain.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A composite function \( (f \circ g)(x) \), which is \( f(g(x)) \), is defined only when the range of the inner function \( g(x) \) is a subset of the domain of the outer function \( f(x) \).
Key Formula or Approach:
The domain of the function \( f(x) = \sin^{-1}x \) is the interval \( [-1, 1] \).
Therefore, for the composite function to exist, we must satisfy:
\[ -1 \leq g(x) \leq 1 \]

Step 2: Detailed Explanation:
Substitute the given expression for \( g(x) = x - 2 \) into the domain condition:
\[ -1 \leq x - 2 \leq 1 \]
To isolate \( x \), add 2 to all parts of the inequality:
\[ -1 + 2 \leq x \leq 1 + 2 \]
\[ 1 \leq x \leq 3 \]
This interval corresponds to the closed set \( [1, 3] \).

Step 3: Final Answer:
The largest domain of \( g(x) \) such that the composite function is defined is \( [1, 3] \).