Question

Let f : R - {3} → R - {1} be defined by f(x) = (x - 2)/(x - 3). Let g : R → R be given as g(x) = 2x - 3. Then, the sum of all the values of x for which f⁻¹(x) + g⁻¹(x) = 13/2 is equal to.

• A. 2
• B. 5
• C. 3
• D. 7

Hint:

To find the inverse of a function, swap $x$ and $y$ and solve for $y$.

Answer 1

The Correct Option is B

Step 1: Find $f^{-1}(x)$. Let $y = \frac{x-2}{x-3} \implies xy - 3y = x - 2 \implies x(y-1) = 3y-2$. $x = \frac{3y-2}{y-1} \implies f^{-1}(x) = \frac{3x-2}{x-1}$.
Step 2: Find $g^{-1}(x)$. Let $y = 2x-3 \implies x = \frac{y+3}{2} \implies g^{-1}(x) = \frac{x+3}{2}$.
Step 3: Solve $\frac{3x-2}{x-1} + \frac{x+3}{2} = \frac{13}{2}$. $\frac{3x-2}{x-1} = \frac{13 - x - 3}{2} = \frac{10-x}{2}$. $2(3x-2) = (x-1)(10-x) \implies 6x - 4 = 10x - x^2 - 10 + x$. $x^2 - 5x + 6 = 0 \implies (x-2)(x-3) = 0$.
Step 4: Values of $x$ are 2 and 3. Sum $= 2+3 = 5$.