Let $f (x) = x^{2}$ $f(x)=x^2$ and $g (x) = \sqrt{x}$ $g(x)=sqrtx$ , how do you find the domain and rules of $(f \cdot g) (x)$ $(f*g)(x)$ ?

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Let $f (x) = x^{2}$ $f(x)=x^2$ and $g (x) = \sqrt{x}$ $g(x)=sqrtx$ , how do you find the domain and rules of $(f \cdot g) (x)$ $(f*g)(x)$ ?

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Answer 1 · 2017-01-31T20:13:33

$f (g (x)) = x, x \geq 0$ $f(g(x)) = x, x>=0$ .

Explanation:

Seeing as this is posted under function composition, I'll treat

$(f \cdot g) (x)$ $(f * g)(x)$ as the composition $f (g (x))$ $f(g(x))$ .

The domain of $f$ $f$ is $RR$ , since any real $x$ can be squared.

The domain of $g$ is $[0, +infty)$ because roots only accept non negative input.

The composition is defined, when $x$ is in the domain of $g$ , and $g(x)$ is in the domain of $f$ . In other words:

With $f(g(x))$ , $x$ is the input (independent variable) for $g$ , and $g(x)$ is the input for $f$ . Therefore, the inputs each must be part of the domains of the functions they are to be used in. We have the following constraints:

$x$ must be non-negative, since it goes into the function $g(x) = sqrtx$ .

$g(x)$ must be real (it obviously is).

Therefore,

$f(g(x)) = (sqrtx)^2 = x$ , for $x >= 0$ .

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Let $f (x) = x^{2}$ $f(x)=x^2$ and $g (x) = \sqrt{x}$ $g(x)=sqrtx$ , how do you find the domain and rules of $(f \cdot g) (x)$ $(f*g)(x)$ ?

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Let $f (x) = x^{2}$ $f(x)=x^2$ and $g (x) = \sqrt{x}$ $g(x)=sqrtx$ , how do you find the domain and rules of $(f \cdot g) (x)$ $(f*g)(x)$ ?