Question

If $f\left(x\right)=-9x^3$ and $g\left(x\right)=-3x$ then choose the correct options
[Given : $(f.g)\left(x\right)=f\left(x\right).g\left(x\right),\left(f/g\right)\left(x\right)=f\left(x\right)/g\left(x\right)$]

• A. $(f.g)(-2)=432$
• B. $\left(f/g\right)(-1/3)=1/3$
• C. $(f.g)(-2)=-427$
• D. $\left(f/g\right)(-1/3)=3$

Answer 1

Answer: (A), (B)

The correct options are
A $\left(f/g\right)(-1/3)=1/3$
B $(f.g)(-2)=432$

(i) We are given that $(f\cdot g)\left(x\right)=f\left(x\right)\cdot g\left(x\right)$ where $f\left(x\right)=-9x^3$ and $g\left(x\right)=-3x$, then we have:

$(f\cdot g)\left(x\right)=\left(f\right)\left(x\right)\cdot \left(g\right)\left(x\right)=(-9x^3)\cdot (-3x)=27x^4$

Therefore, $(f\cdot g)\left(x\right)=27x^4$

Now, let us substitute $x=-2$ as shown below:

$(f\cdot g)(-2)=27{(-2)}^4=27\times 16=432$

Hence, $(f\cdot g)(-2)=432$

(ii) We are also given that $\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$ where $f\left(x\right)=-9x^3$ and $g\left(x\right)=-3x$, then we have:

$\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}=\frac{-9x^3}{-3x}=3x^2$

Therefore, $\left(\frac{f}{g}\right)\left(x\right)=3x^2$

Now, let us substitute $x=-\frac{1}{3}$ as shown below:

$\left(\frac{f}{g}\right)(-\frac{1}{3})=3{(-\frac{1}{3})}^2=3\times \frac{1}{9}=\frac{1}{3}$

Hence, $\left(\frac{f}{g}\right)(-\frac{1}{3})=\frac{1}{3}$