Consider the following functions from R to R, where R is the set of real numbers:
(A) \(f(x) = |x|\)
(B) \(f(x) = 2^{x}\)
(C) \(f(x) = x^{2} + 2x\)
(D) \(f(x) = 5x + 4\)
Which of the above functions are one-one?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B) and (D) only
A function is one-one when different inputs always give different outputs. One pair of different inputs with the same output is enough to show a function is not one-one.
For \(|x|\):
\(|-3| = 3 = |3|\)
For \(x^{2} + 2x\):
\(f(0) = 0\)
\(f(-2) = 4 - 4 = 0\)
For \(2^{x}\): the function is strictly increasing, so \(2^{a} = 2^{b}\) gives \(a = b\).
For \(5x + 4\):
\(5a + 4 = 5b + 4\)
\(a = b\)
So \(2^{x}\) and \(5x + 4\) are one-one, while \(|x|\) and \(x^{2} + 2x\) are not.
Checking the statements
(A) \(f(x) = |x|\) — Incorrect. \(f(-3) = f(3) = 3\).
(B) \(f(x) = 2^{x}\) — Correct. Strictly increasing, so one-one.
(C) \(f(x) = x^{2} + 2x\) — Incorrect. \(f(0) = f(-2) = 0\).
(D) \(f(x) = 5x + 4\) — Correct. \(5a + 4 = 5b + 4\) gives \(a = b\).