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?

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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?

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Answer 1 · 2015-07-04T13:57:47

See below.

Explanation:

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

For $f (x) = \sqrt{2} x + 3$ $f(x) = sqrt2 x+3$ , the domain is all real numbers. $(- \infty, \infty)$ $(-oo, oo)$

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

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.

$(f + g) (x) = f (x) + g (x)$ $(f+g)(x) = f(x)+g(x)$

$= (\sqrt{2} x + 3) + (x^{2} - 2)$ $= (sqrt2 x+3) + (x^2-2)$
$= x^{2} + \sqrt{2} x + 1$ $= x^2+sqrt2 x +1$

$(f - g) (x) = f (x) - g (x)$ $(f-g)(x) = f(x)-g(x)$

$= (\sqrt{2} x + 3) - (x^{2} - 2)$ $= (sqrt2 x+3) - (x^2-2)$
$= \sqrt{2} x + 3 - x^{2} + 2$ $= sqrt2 x +3 -x^2 +2$
$= - x^{2} + \sqrt{2} x + 5$ $= -x^2+sqrt2 x + 5$

$(\frac{f}{g}) (x) = \frac{f (x)}{g (x)}$ $(f/g)(x) = f(x)/g(x)$

$= \frac{\sqrt{2} x + 3}{x^{2} - 2}$ $= (sqrt2 x+3) / (x^2-2)$ (For $x \neq \pm \sqrt{2}$ $x != +- sqrt2$ .)

$(\frac{g}{f}) (x) = \frac{g (x)}{f (x)}$ $(g/f)(x) = g(x)/f(x)$

$= \frac{x^{2} - 2}{\sqrt{2} x + 3}$ $= (x^2-2)/(sqrt2 x+3)$ (For $x \neq - \frac{3}{\sqrt{2}}$ $x != - 3/sqrt2$ .)

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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?

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Explanation:

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