Let \(f : [2, 5] \to [a, b]\) be defined by \(f(x) = 3x - 1\). If f is one-one and onto, then \(a + b\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 19
Since \(f(x) = 3x - 1\) is increasing, its smallest value is at \(x = 2\) and its largest at \(x = 5\):
\(f(2) = 3 \times 2 - 1 = 5\)
\(f(5) = 3 \times 5 - 1 = 14\)
So the range of f is \([5, 14]\). For f to be onto, the codomain must equal the range:
\(a = 5\) and \(b = 14\)
f is also one-one, since \(3p - 1 = 3q - 1\) gives \(p = q\). Hence
\(a + b = 5 + 14 = 19\)