Set, Relations and FunctionsICAI SM, MTP May 20Question 1974 of 217
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If f(x)=x1−x\displaystyle f(x) = \frac{x}{1-x} & g(x)=x1+x\displaystyle g(x) = \frac{x}{1+x}, then gof(x)\displaystyle gof(x) is

Options

Ax1−x\displaystyle \frac{x}{1-x}
Bx1+x\displaystyle \frac{x}{1+x}
Cx\displaystyle x
DNone of these
For any discrepancies in this question, email contact@cadada.in

Correct Answer

✅ Option b — x1+x\displaystyle \frac{x}{1+x}

All Options:

  • Ax1−x\displaystyle \frac{x}{1-x}
  • Bx1+x\displaystyle \frac{x}{1+x}
  • Cx\displaystyle x
  • DNone of these

Detailed Solution & Explanation

Let us find the composition of the two functions g(f(x))\displaystyle g(f(x)):

Given:
f(x)=x1−xf(x) = \frac{x}{1-x}
g(x)=x1+xg(x) = \frac{x}{1+x}

We want to find g(f(x))\displaystyle g(f(x)) (which is (g∘f)(x)\displaystyle (g \circ f)(x)):
g(f(x))=g(x1−x)g(f(x)) = g\left(\frac{x}{1-x}\right)
Substitute x1−x\displaystyle \frac{x}{1-x} into the function g(x)\displaystyle g(x) in place of x\displaystyle x:
g(f(x))=x1−x1+x1−xg(f(x)) = \frac{\frac{x}{1-x}}{1 + \frac{x}{1-x}}
To simplify this complex fraction, let us find a common denominator for the terms in the denominator:
1+x1−x=1−x1−x+x1−x=(1−x)+x1−x=11−x1 + \frac{x}{1-x} = \frac{1-x}{1-x} + \frac{x}{1-x} = \frac{(1-x) + x}{1-x} = \frac{1}{1-x}
Now, substitute this simplified denominator back into our expression for g(f(x))\displaystyle g(f(x)):
g(f(x))=x1−x11−xg(f(x)) = \frac{\frac{x}{1-x}}{\frac{1}{1-x}}
Multiply the numerator by the reciprocal of the denominator:
g(f(x))=x1−x⋅1−x1=xg(f(x)) = \frac{x}{1-x} \cdot \frac{1-x}{1} = x

**Discrepancy & Typographical Error:**
Mathematically, the composite function (g∘f)(x)\displaystyle (g \circ f)(x) simplifies to x\displaystyle x, which corresponds to **Option C**. However, some textbook answer keys / mock test papers (MTP May 20) contain a typographical error where the answer is miskeyed as **Option B** (x1+x\displaystyle \frac{x}{1+x}). We have proven that the true mathematical value of the expression is x\displaystyle x (Option C), but we acknowledge the book's key discrepancy.

Hence, **Option B** is the correct answer (as per the textbook key), but the mathematically proven correct value is x\displaystyle x (**Option C**).

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