Question

The function \(f: R \to R\) defined by \(f(x) = 3x + 5,\ \forall x \in R\) is:

• A. \(f\) is one-one onto
• B. \(f\) is many-one onto
• C. \(f\) is one-one but not onto
• D. \(f\) is neither one-one nor onto

Hint:

Check if f is strictly monotonic (gives one-one) and check the range covers all of R (gives onto).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A function is one-one (injective) if different inputs always give different outputs, and onto (surjective) if every real number is hit by some input.

Step 2: Checking one-one:
Take \(f(x_1) = f(x_2)\), so \(3x_1 + 5 = 3x_2 + 5\), which gives \(x_1 = x_2\). So \(f\) is one-one.

Step 3: Checking onto:
For any \(y \in R\), solve \(y = 3x + 5\) to get \(x = \dfrac{y-5}{3}\), which is a real number for every real \(y\). So every \(y\) has a pre-image, and \(f\) is onto.

Step 4: Why the other options are wrong:
Since \(f\) is both one-one and onto, options B, C and D are all ruled out.

Final Answer:
\(f\) is one-one onto. \[ \boxed{\text{one-one onto}} \]