Let f(x) = 2x + 5 and g(x) = x² + x. Describe (i) f+g (ii) f−g (iii) fg (iv) f/g and find the domain in each case.
Solution
Given:
\[
f(x)=2x+5
\]
\[
g(x)=x^2+x
\]
(i) Find \(f+g\)
\[
(f+g)(x)=f(x)+g(x)
\]
\[
=(2x+5)+(x^2+x)
\]
\[
=x^2+3x+5
\]
Domain: Both functions are defined for all real numbers.
\[
\text{Domain of } (f+g)=\mathbb{R}
\]
(ii) Find \(f-g\)
\[
(f-g)(x)=f(x)-g(x)
\]
\[
=(2x+5)-(x^2+x)
\]
\[
=-x^2+x+5
\]
Domain:
\[
\text{Domain of } (f-g)=\mathbb{R}
\]
(iii) Find \(fg\)
\[
(fg)(x)=f(x)\cdot g(x)
\]
\[
=(2x+5)(x^2+x)
\]
\[
=2x^3+2x^2+5x^2+5x
\]
\[
=2x^3+7x^2+5x
\]
Domain:
\[
\text{Domain of } (fg)=\mathbb{R}
\]
(iv) Find \(f/g\)
\[
\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}
\]
\[
=\frac{2x+5}{x^2+x}
\]
\[
=\frac{2x+5}{x(x+1)}
\]
For the quotient function, denominator should not be zero.
\[
x(x+1)\neq 0
\]
\[
x\neq 0,\,-1
\]
Domain:
\[
\text{Domain of } \left(\frac{f}{g}\right)=\mathbb{R}-\{0,-1\}
\]