Question

Let f(x) = 2x + 5 and g(x) = x² + x. Describe (i) f + g (ii) f − g (iii) fg (iv) f/g. Find the domain in each case.

Answer 1

Given:
f(x) = 2x + 5 and g(x) = x² + x
Clearly, f (x) and g (x) assume real values for all x.
Hence,
domain (f) = R and domain (g) = R.
$\therefore D\left(f\right)\cap D\left(g\right)=R$ .

Now,
(i) (f + g) : R → R is given by (f + g) (x) = f (x) + g (x) = 2x + 5 + x² + x = x² + 3x + 5.
Hence, domain ( f + g) = R .

(ii) (f $-$ g) : R → R is given by (f $-$ g) (x) = f (x) $-$ g (x) = (2x + 5) $-$ (x² + x) = 5 + x $-$ x²
Hence, domain ( f $-$ g) = R.

(iii) (fg) : R → R is given by (fg) (x) = f(x).g(x) = (2x + 5)(x² + x)
= 2x³ + 2x² + 5x² + 5x
= 2x³ + 7x² + 5x
Hence, domain ( f.g) = R .

(iv) Given:
g(x) = x² + x
g(x) = 0 ⇒ x² + x = 0 = x(x+ 1) = 0
⇒ x = 0 or (x + 1) = 0
⇒ x = 0 or x = $-$ 1
Now,
$\frac{f}{g}:R-\left\{-1,0\right\}\to R\mathrm{is}\mathrm{given}\mathrm{by}\left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(\mathrm{x}\right)=\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}=\frac{2\mathrm{x}+5}{{\mathrm{x}}^2+\mathrm{x}}$ .
Hence, $\text{d}\mathrm{omain}\left(\frac{\mathrm{f}}{\mathrm{g}}\right)=\mathrm{R}-\left\{-1,0\right\}$.