# Find (f+g)(x) , (f-g)(x) , (f/g)(x) and (g/f)(x) for the following f and g . State the domain for f(x)=(√2x+3) ; g(x)=x²-2?

Jul 4, 2015

See below.

#### Explanation:

State the domain for f(x)=(√2x+3) ; g(x)=x²-2.

For $f \left(x\right) = \sqrt{2} x + 3$, the domain is all real numbers. $\left(- \infty , \infty\right)$

Also for $g \left(x\right) = {x}^{2} - 2$, the domain is all real numbers. $\left(- \infty , \infty\right)$.

Neither function involves the possibility of dividing by zeros or of trying to find an even root of a negative number. Nor of any other undefined expression.

$\left(f + g\right) \left(x\right) = f \left(x\right) + g \left(x\right)$

$= \left(\sqrt{2} x + 3\right) + \left({x}^{2} - 2\right)$
$= {x}^{2} + \sqrt{2} x + 1$

$\left(f - g\right) \left(x\right) = f \left(x\right) - g \left(x\right)$

$= \left(\sqrt{2} x + 3\right) - \left({x}^{2} - 2\right)$
$= \sqrt{2} x + 3 - {x}^{2} + 2$
$= - {x}^{2} + \sqrt{2} x + 5$

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

$= \frac{\sqrt{2} x + 3}{{x}^{2} - 2}$ (For $x \ne \pm \sqrt{2}$.)

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

$= \frac{{x}^{2} - 2}{\sqrt{2} x + 3}$ (For $x \ne - \frac{3}{\sqrt{2}}$.)