GATE CS 2015 Set 1 Q5 — Sets, Relations and Functions

We are given the functions:

$$g(x) = 1 - x$$

$$h(x) = \frac{x}{x-1}$$

Let us compute the numerator and denominator of the given expression step-by-step.

Step 1: Compute the Numerator $g(h(x))$

$$g(h(x)) = 1 - h(x) = 1 - \frac{x}{x-1}$$

$$g(h(x)) = \frac{(x-1) - x}{x-1} = \frac{-1}{x-1} = \frac{1}{1-x}$$

Step 2: Compute the Denominator $h(g(x))$

$$h(g(x)) = \frac{g(x)}{g(x)-1} = \frac{1-x}{(1-x)-1}$$

$$h(g(x)) = \frac{1-x}{-x} = \frac{x-1}{x}$$

Step 3: Compute the Ratio $\frac{g(h(x))}{h(g(x))}$

$$\frac{g(h(x))}{h(g(x))} = \frac{\frac{1}{1-x}}{\frac{x-1}{x}} = \frac{1}{1-x} \cdot \frac{x}{x-1} = \frac{x}{(1-x)(x-1)}$$

Step 4: Check Option A

Let us evaluate Option A, which is $\frac{h(x)}{g(x)}$:

$$\frac{h(x)}{g(x)} = \frac{\frac{x}{x-1}}{1-x} = \frac{x}{(x-1)(1-x)}$$

Since this matches the ratio computed in Step 3, the correct option is A.