Functions
Composition of functions
nta_pyq_2025_apr
Grade 12
Question:
Let $f, g : (1, \infty) \to \mathbb{R}$ be defined as $f(x) = \frac{2x+3}{5x+2}$ and $g(x) = \frac{2-3x}{1-x}$. If the range of the function $f \circ g : [2, 4] \to \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to
Step-by-Step Solution
Key Concept: Convert the condition into algebraic equations and simplify the resulting expression.
$f \circ g(x) = f(g(x)) = f\left(\frac{2-3x}{1-x}\right) = \frac{2 \cdot \frac{2-3x}{1-x} + 3}{5 \cdot \frac{2-3x}{1-x} + 2}$
$= \frac{\frac{2(2-3x) + 3(1-x)}{1-x}}{\frac{5(2-3x) + 2(1-x)}{1-x}} = \frac{4 - 6x + 3 - 3x}{10 - 15x + 2 - 2x} = \frac{7 - 9x}{12 - 17x}$
Note: $12 - 17x \neq 0$ for $x \in [2, 4]$
$f \circ g(2) = \frac{7 - 9(2)}{12 - 17(2)} = \frac{7 - 18}{12 - 34} = \frac{-11}{-22} = \frac{1}{2}$
$f \circ g(4) = \frac{7 - 9(4)}{12 - 17(4)} = \frac{7 - 36}{12 - 68} = \frac{-29}{-56} = \frac{29}{56}$
Range of $f \circ g$: $[\alpha, \beta] = \left[\frac{1}{2}, \frac{29}{56}\right]$
$\beta - \alpha = \frac{29}{56} - \frac{1}{2} = \frac{29}{56} - \frac{28}{56} = \frac{1}{56}$
$\therefore \frac{1}{\beta - \alpha} = \frac{1}{\frac{1}{56}} = 56$
Correct Answer: 4