Let \(f : R \to B\) be defined by \(f(x) = 4 - 3\sin x\), where R is the set of real numbers. If f is onto, then B is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \([1, 7]\)
A function is onto exactly when its codomain equals its range. So B must be the range of f.
For every real x,
\(-1 \le \sin x \le 1\)
\(-3 \le 3\sin x \le 3\)
\(-3 \le -3\sin x \le 3\)
\(1 \le 4 - 3\sin x \le 7\)
Both ends are reached, since \(\sin x\) takes every value from \(-1\) to 1:
\(\sin x = 1\) gives \(f(x) = 1\)
\(\sin x = -1\) gives \(f(x) = 7\)
So the range is \([1, 7]\), and \(B = [1, 7]\).