Question

If $f$ and g are two real valued functions defined as

$f\left(x\right)=2x+1,g\left(x\right)=x^2+1$, then find:

(i) $f+g$

(ii) $f-g$

(iii) $fg$

(iv) $\frac{f}{g}$

Answer 1

$f\left(x\right)=2x+1,g\left(x\right)=x^2+1,andg\left(x\right)=x^2+1$

Find the addition of given functions
$(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)$
= $2x+1+x^2+1$
= $x^2+2x+2$
Hence, $(f+g)\left(x\right)=x^2+2x+2$

(ii)

$f\left(x\right)=2x+1,andg\left(x\right)=x^2+1,andg\left(x\right)=x^2+1$

Find the Substraction of given functions
$(f-g)\left(x\right)=f\left(x\right)-g\left(x\right)$
= $2x+1-(x^2+1)$
= $2x+1-x^2-1$
=$2x-x^2$
Hence, $(f-g)\left(x\right)=2x-x^2$

(iii)
$f\left(x\right)=2x+1andg\left(x\right)=x^2+1$

Find the addition of given functions
$\left(fg\right)\left(x\right)=f\left(x\right).g\left(x\right)$
= $2x+1\left)\right(x^2+1)$
= $2x^3+x^2+2x+1$
Hence, $\left(fg\right)\left(x\right)=2x^3+x^2+2x+1$

(iv)

$f\left(x\right)=2x+1andg\left(x\right)=x^2+1$
Find quotient of given functions

$\left(\frac{f}{g}\right)\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

=$\left(\frac{2x+1}{x^2+1}\right)$

Hence,$\left(\frac{f}{g}\right)\left(x\right)=\left(\frac{2x+1}{x^2+1}\right)$