Other Properties

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1

Define the functions and on the set of real numbers as follows:

$f (x) = \sqrt{x- 16}$

$g(x) = \sqrt{x- 4}$

Give the natural domain of the composite function $f \circ g$ .

$[260, \infty)$

$[4, \infty)$

$[16, \infty)$

$[32, \infty)$

$[0 , \infty)$

Explanation

The natural domain of the composite function $f \circ g$ is defined to be the intersection of the two sets.

One set is the natural domain of . Since is defined to be the square root of an expression, the radicand must be nonnegative. Therefore,

$x - 4 \geq 0$

$x \geq 4$

This set is $[4, \infty)$ .

The other set is the set of numbers that the function pairs with a number within the domain of . Since the radicand of the square root in must be nonnegative,

$x - 16 \geq 0$

$x \geq 16$

For to fall within this set:

$g(x) \geq 16$

$\sqrt{x- 4} \geq 16$

$x - 4 \geq 256$

$x \geq 260$

This set is $[260, \infty)$ .

$[4, \infty) \cap [260, \infty) = [260, \infty)$ , which is the natural domain.

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