Let f(x) and g(x) be two functions such that g(x)=x-(1)/(x) and f o g(x)=x^3-(1)/(x^3) . What is g[f(x)-3 x] equal to?

Correct answer: A. x^3-(1)/(x^3) f(x) and g(x) are two functions aligned g(x) & =x-(1)/(x) and f o g(x)=x^3-(1)/(x^3) f(x-(1)/(x)) & =x^3-(1)/(x^3) f(x-(1)/(x)) & =(x-(1)/(x))^3+3(x-(1)/(x)) aligned f(x)=x^3+3 x ....(i) aligned g[f(x)-3 x] & =f(x)-3 x-(1)/(f(x)-3 x) g(f(x)-3 x] & =x^3+3 x-3 x-(1)/(x^3+3 x-3 x) & =x^3-(1)/(x^3) g(f(x)-3 x] & =x^3-(1)/(x^3) aligned

NDA 2024

Exam: NDA
Year: 2024
Subject: Mathematics
Chapter: Functions

NDA 2024 · Mathematics · Functions

Let f(x) and g(x) be two functions such that g(x)=x-(1)/(x) and f o...

Let

f(x)

and

g(x)

be two functions such that

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

and

f

g(x)=x^3-\frac{1}{x^3}

. What is

g[f(x)-3 x]

equal to?

Options

A $x^3-\frac{1}{x^3}$
B $x^3+\frac{1}{x^3}$
C $x^2-\frac{1}{x^2}$
D $x^2+\frac{1}{x^2}$

Correct answer

A. $x^3-\frac{1}{x^3}$

Step-by-step solution

\because f(x)

and g(x) are two functions

\begin{aligned} g(x) & =x-\frac{1}{x} \text{ and } f o g(x)=x^3-\frac{1}{x^3} \\ f\left(x-\frac{1}{x}\right) & =x^3-\frac{1}{x^3} \\ \Rightarrow f\left(x-\frac{1}{x}\right) & =\left(x-\frac{1}{x}\right)^3+3\left(x-\frac{1}{x}\right) \end{aligned}

f(x)=x^3+3 x \quad ....(i)

\begin{aligned} g[f(x)-3 x] & =f(x)-3 x-\frac{1}{f(x)-3 x} \\ g(f(x)-3 x] & =x^3+3 x-3 x-\frac{1}{x^3+3 x-3 x} \\ & =x^3-\frac{1}{x^3} \\ g(f(x)-3 x] & =x^3-\frac{1}{x^3}\end{aligned}

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