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:
The composite function \((f \circ g)(x) = f(g(x))\) is defined only when the range of \(g(x)\) falls within the domain of \(f(x)\).
The domain of the inverse sine function is restricted.
Step 2: Key Formula or Approach:
The domain of \(f(y) = \sin^{-1}(y)\) is \([-1, 1]\).
Set the argument \(g(x)\) such that \(-1 \leq g(x) \leq 1\).
Step 3: Detailed Explanation:
We have the composite function:
\[ f(g(x)) = \sin^{-1}(x - 2) \] For this to be real and defined, the argument must be bounded:
\[ -1 \leq x - 2 \leq 1 \] Add 2 to all parts of the inequality:
\[ -1 + 2 \leq x \leq 1 + 2 \] \[ 1 \leq x \leq 3 \] Thus, the domain of \(x\) is \([1, 3]\).
Step 4: Final Answer:
The largest domain is \([1, 3]\).