Question

The function defined by $f(x) = \frac{2x+3}{3x+4}, x \neq -\frac{4}{3}$ is

• A. only one one
• B. only onto
• C. onto for $y \neq \frac{2}{3}$ and one-one
• D. neither one-one nor onto

Hint:

For $y = \frac{ax+b}{cx+d}$, the value excluded from the range is always the ratio of the coefficients of $x$, i.e., $y = a/c$.

Answer 1

The Correct Option is C

Step 1: Concept
A linear fractional function $f(x) = \frac{ax+b}{cx+d}$ is always one-one if $ad - bc \neq 0$.

Step 2: Meaning
For one-one: Let $f(x_1) = f(x_2)$ and solve for $x_1 = x_2$. For onto: Find the range by expressing $x$ in terms of $y$.

Step 3: Analysis
1. One-one: $ad-bc = (2)(4) - (3)(3) = 8-9 = -1 \neq 0$. Thus, it is one-one. 2. Onto: Let $y = \frac{2x+3}{3x+4} \implies 3xy + 4y = 2x + 3 \implies x(3y-2) = 3-4y \implies x = \frac{3-4y}{3y-2}$. $x$ is defined for all $y$ except $3y-2 = 0 \implies y = 2/3$.

Step 4: Conclusion
The function is one-one and onto for its range (all real numbers except 2/3). Final Answer: (C)