NDA 1 2024 Maths, question 85

Topic: Differential calculus

Passage · Q85 to Q86

Let $f(x)$ and $g(x)$ be two functions such that $g(x)=x-\frac{1}{x}$ and $f\circ g(x)=x^{3}-\frac{1}{x^{3}}$ .

Q85

What is $g[f(x)-3x]$ equal to ?

(a) $x^{3}-\frac{1}{x^{3}}$ Correct answer
(b) $x^{3}+\frac{1}{x^{3}}$
(c) $x^{2}-\frac{1}{x^{2}}$
(d) $x^{2}+\frac{1}{x^{2}}$

Solution.

Answer: (a) $x^{3}-\frac{1}{x^{3}}$

If t = x − 1/x, then x³ − 1/x³ = t³ + 3t.
Thus f(t) = t³ + 3t, and f(x) − 3x = x³.
Applying g gives x³ − 1/x³, for x ≠ 0 as required by g.