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QuestionGiven functions f(x)=xf(x)=\sqrt{x} and g(x)=x+3g(x)=x+3, find (a)(f∘g)(x)(a)(f \circ g)(x) and its domain, and (b)(g∘f)(x)(b)(g \circ f)(x) and its domain.

Studdy Solution

STEP 1

Assumptions1. We have two functions, f(x)=xf(x) = \sqrt{x} and g(x)=x+3g(x) = x +3. . We need to find (f∘g)(x)(f \circ g)(x) and its domain, and (g∘f)(x)(g \circ f)(x) and its domain.

STEP 2

First, let's find (f∘g)(x)(f \circ g)(x). The notation (f∘g)(x)(f \circ g)(x) means f(g(x))f(g(x)). So we need to substitute g(x)g(x) into f(x)f(x).
(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

STEP 3

Now, substitute g(x)g(x) into f(x)f(x).
(f∘g)(x)=f(x+3)(f \circ g)(x) = f(x +3)

STEP 4

Substitute x+3x +3 into f(x)f(x).
(f∘g)(x)=x+3(f \circ g)(x) = \sqrt{x +3}

STEP 5

Now, let's find the domain of (f∘g)(x)(f \circ g)(x). The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For the square root function, the input must be greater than or equal to0.
x+3≥0x +3 \geq0

STEP 6

olve the inequality to find the domain.
x≥−3x \geq -3So the domain of (f∘g)(x)(f \circ g)(x) is [−3,∞)[-3, \infty).

STEP 7

Now, let's find (g∘f)(x)(g \circ f)(x). The notation (g∘f)(x)(g \circ f)(x) means g(f(x))g(f(x)). So we need to substitute f(x)f(x) into g(x)g(x).
(g∘f)(x)=g(f(x))(g \circ f)(x) = g(f(x))

STEP 8

Now, substitute f(x)f(x) into g(x)g(x).
(g∘f)(x)=g(x)(g \circ f)(x) = g(\sqrt{x})

STEP 9

Substitute x\sqrt{x} into g(x)g(x).
(g∘f)(x)=x+3(g \circ f)(x) = \sqrt{x} +3

STEP 10

Now, let's find the domain of (g∘f)(x)(g \circ f)(x). The domain of a function is the set of all possible input values (x-values) which will produce a valid output. For the square root function, the input must be greater than or equal to0.
x≥0x \geq0So the domain of (g∘f)(x)(g \circ f)(x) is [0,∞)[0, \infty).
In conclusion,(a) (f∘g)(x)=x+3(f \circ g)(x) = \sqrt{x +3} with domain [−3,∞)[-3, \infty), and(b) (g∘f)(x)=x+3(g \circ f)(x) = \sqrt{x} +3 with domain [0,∞)[0, \infty).

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