Question

If $f$ and $g$ are two functions such that $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)$ for all $x$. Then $f$ and $g$ may be defined as

• A. $f\left(x\right)=\sqrt{x},g\left(x\right)=\cos x$
• B. $f\left(x\right)=x^3,g\left(x\right)=x+1$
• C. $f\left(x\right)=x-1,g\left(x\right)=x^2+1$
• D. $f\left(x\right)=x^m,g\left(x\right)=x^n$ where $m,n$ are unequal integers

Answer 1

The correct option is D $f\left(x\right)=x^m,g\left(x\right)=x^n$ where $m,n$ are unequal integers

A$\left)f\right(x)=$$\sqrt{x},g\left(x\right)=cosxthen\left(fg\right)\left(x\right)=\sqrt{\cos x}and\left(gf\right)\left(x\right)=\cos \sqrt{x}$.They are not equal.

B)$f\left(x\right)=x^3,g\left(x\right)=x+1then\left(fg\right)\left(x\right)=(x+1{)}^{3}and(gf\left)\right(x)=x^3+1$.They are not equal.

C)$f\left(x\right)=x1,g\left(x\right)=x^2+1then\left(fg\right)\left(x\right)=x^{2}and\left(gf\right)\left(x\right)=x^2-2x$. They are not equal.

D)$f\left(x\right)=x^m,g\left(x\right)=x^{n}then\left(fg\right)\left(x\right)=x^{n+m}and\left(gf\right)\left(x\right)=x^{n+m}$.Hence $\left(fg\right)\left(x\right)=\left(gf\right)\left(x\right)$