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: Test with the definition directly:
A linear function $f(x) = mx + c$ with $m \neq 0$ is always a bijection from $R$ to $R$, since it is strictly monotonic and unbounded on both sides.

Step 2: Strictly increasing check:
Here $m = 3 > 0$, so $f$ is strictly increasing on $R$; a strictly increasing function can never repeat a value, so it is automatically one-one.

Step 3: Range check:
As $x \to -\infty$, $f(x) \to -\infty$, and as $x \to \infty$, $f(x) \to \infty$. A continuous function going from $-\infty$ to $\infty$ covers every real value, so the range is all of $R$, meaning $f$ is onto.

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