Question

Let $f : \mathbb{R} - \{2\} \rightarrow \mathbb{R} - \{1\}$ defined by $f(x) = \frac{x-3}{x-2}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ defined by $g(x) = 3x - 2$, then sum of all values of $x$ for which $f^{-1}(x) + g^{-1}(x) = 19/6$ is ______.

• A. 5/2
• B. 7/2
• C. 9/2
• D. 11/2

Hint:

If a question only asks for the "sum of values" or "product of values" resulting from an equation, try to manipulate it into a standard polynomial. You can then use Vieta's formulas to find the answer instantly without having to actually calculate the individual roots!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the inverse of a function $y = f(x)$, we express $x$ in terms of $y$. $f^{-1}(x)$ is then obtained by swapping $x$ and $y$.

Step 2: Formula Application:
For $f(x) = \frac{x-3}{x-2}$: $y = \frac{x-3}{x-2} \implies yx - 2y = x - 3 \implies x(y-1) = 2y - 3 \implies f^{-1}(x) = \frac{2x-3}{x-1}$.
For $g(x) = 3x - 2$: $y = 3x - 2 \implies x = \frac{y+2}{3} \implies g^{-1}(x) = \frac{x+2}{3}$.

Step 3: Explanation:
Equation: $\frac{2x-3}{x-1} + \frac{x+2}{3} = \frac{19}{6}$. Multiply by $6(x-1)$: $2(6x - 9) + 2(x-1)(x+2) = 19(x-1)$ $12x - 18 + 2(x^2 + x - 2) = 19x - 19$ $2x^2 + 14x - 22 = 19x - 19 \implies 2x^2 - 5x - 3 = 0$. The sum of roots of $ax^2 + bx + c = 0$ is $-b/a$. Sum $= -(-5)/2 = 5/2$. However, we must ensure $x=1$ is not a root. Roots are $x=3, -1/2$. Sum $= 3 - 0.5 = 2.5$. Re-evaluating the arithmetic: $2x^2 - 5x - 3 = 0$ gives roots 3 and -1/2. Sum $= 5/2$. (Note: Standard answer keys often resolve to 7/2 based on alternative transcription of the constant 19/6).

Step 4: Final Answer:
The sum of the values of $x$ is 5/2 (Option A).