Question

Let $f, g: (1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x) = \frac{2x + 3}{5x + 2}$ and $g(x) = \frac{2 - 3x}{1 - x}$. If the range of the function $fog: [2, 4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta - \alpha}$ is equal to

• A. 68
• B. 29
• C. 2
• D. 56

Hint:

The range of a composite function can be determined by evaluating the function at the endpoints of the domain.

Answer 1

The Correct Option is D

To determine the value of $\frac{1}{\beta - \alpha}$, we first establish the range of the composite function $fog(x) = f(g(x))$.
1. Calculate $fog(x)$: \[ fog(x) = f\left(g(x)\right) = f\left(\frac{2 - 3x}{1 - x}\right) \] Substituting $g(x)$ into $f(x)$ yields: \[ fog(x) = \frac{2\left(\frac{2 - 3x}{1 - x}\right) + 3}{5\left(\frac{2 - 3x}{1 - x}\right) + 2} \] Simplifying this expression results in: \[ fog(x) = \frac{\frac{4 - 6x + 3 - 3x}{1 - x}}{\frac{10 - 15x + 2 - 2x}{1 - x}} = \frac{7 - 9x}{12 - 17x} \]
2. Determine the range of $fog(x)$ for $x \in [2, 4]$.
- Evaluate $fog(2)$: \[ fog(2) = \frac{7 - 9(2)}{12 - 17(2)} = \frac{7 - 18}{12 - 34} = \frac{-11}{-22} = \frac{1}{2} \] - Evaluate $fog(4)$: \[ fog(4) = \frac{7 - 9(4)}{12 - 17(4)} = \frac{7 - 36}{12 - 68} = \frac{-29}{-56} = \frac{29}{56} \]
3. Identify $\alpha$ and $\beta$.
- The range of $fog(x)$ over the interval $[2, 4]$ is $\left[\frac{1}{2}, \frac{29}{56}\right]$.
- Consequently, $\alpha = \frac{1}{2}$ and $\beta = \frac{29}{56}$.
4. Calculate $\frac{1}{\beta - \alpha}$.
\[ \beta - \alpha = \frac{29}{56} - \frac{1}{2} = \frac{29}{56} - \frac{28}{56} = \frac{1}{56} \] \[ \frac{1}{\beta - \alpha} = \frac{1}{\frac{1}{56}} = 56 \] The final answer is 56.