Question

If a function \( f: \mathbb{R} - \{l\} \to \mathbb{R} - \{m\} \) defined by \( f(x) = \frac{x+3}{x-2} \) is a bijection, then \( 3l + 2m = \) ?

• A. 10
• B. 12
• C. 8
• D. 14

Hint:

To determine domain and range exclusions in rational functions, look for values that make the denominator zero or the function non-invertible. These give the excluded values \( l \) and \( m \).

Answer 1

The Correct Option is C

The function \( f(x) = \frac{x+3}{x-2} \) is undefined when the denominator is zero, i.e., \[ x - 2 = 0 \Rightarrow x = 2 \] So, the domain excludes \( l = 2 \). Now, find the value which is never attained by the function \( f(x) \). That is, we solve for \( y = \frac{x+3}{x-2} \) and find the value of \( y \) for which the function is undefined or not surjective. Cross-multiplying: \[ y(x - 2) = x + 3 \Rightarrow yx - 2y = x + 3 \Rightarrow yx - x = 2y + 3 \Rightarrow x(y - 1) = 2y + 3 \] This equation fails when \( y = 1 \), as it makes the denominator zero and no solution exists for \( x \). So, \( y = 1 \) is not in the range. Thus, the range excludes \( m = 1 \). Now, compute: \[ 3l + 2m = 3(2) + 2(1) = 6 + 2 = 8 \]