Question

The function \(f(x)=2x\), \(x\in R\) is:

• A. one-one but not onto
• B. one-one and onto
• C. many-one and onto
• D. many-one but not onto

Hint:

Check if \(2x_1=2x_2\) forces \(x_1=x_2\), and if every real y has a pre-image \(y/2\).

Answer 1

The Correct Option is B

Step 1: Use the graph of f(x) = 2x:
The graph of $f(x)=2x$ is a straight line through the origin with slope 2, extending over all real x.

Step 2: Read one-one from the graph:
A strictly increasing straight line never repeats a y-value for two different x-values, so no horizontal line cuts the graph more than once, confirming f is one-one.

Step 3: Read onto from the graph:
As x ranges over all of R, the line extends from $-\infty$ to $+\infty$ on the y-axis with no gaps, so every real value of y is hit by some x, meaning the range equals the codomain R.

Step 4: Conclude the classification:
A graph that passes the horizontal line test everywhere and covers the entire y-axis represents a bijective function.

Final Answer:
The straight-line graph confirms f is both one-one and onto, same as the algebraic check. \[ \boxed{\text{one-one and onto}} \]