Question

use the functions below. - ( f(x) = 2x - 1 ) - ( g(x) = |x + 2| + 3 ) what is ( (f circ g)(x) )? what is the domain of ( (f circ g)(x) )? (1 point) ( circ ) ( (f circ g)(x) = 2|x + 2| + 5 ) and the domain of ( (f circ g)(x) ) is ( x geq -2 ). ( odot ) ( (f circ g)(x) = 2|x + 2| + 5 ) and the domain of ( (f circ g)(x) ) is ( (-infty, infty) ). ( circ ) ( (f circ g)(x) = |2x + 1| + 3 ) and the domain of ( (f circ g)(x) ) is ( (-infty, infty) ). ( circ ) ( (f circ g)(x) = |2x + 1| + 3 ) and the domain of ( (f circ g)(x) ) is ( x geq -2 )

Explanation:

Step1: Recall the definition of function composition

The composition of two functions \( (f \circ g)(x) \) is defined as \( f(g(x)) \). So we need to substitute \( g(x) \) into \( f(x) \).

Given \( f(x) = 2x - 1 \) and \( g(x)=|x + 2|+3 \), we substitute \( g(x) \) into \( f \):

\( (f \circ g)(x)=f(g(x)) = 2(g(x))-1 \)

Step2: Substitute \( g(x) \) into the formula for \( f \)

Substitute \( g(x)=|x + 2|+3 \) into \( 2(g(x))-1 \):

\( 2(|x + 2|+3)-1 \)

Step3: Simplify the expression

First, distribute the 2: \( 2|x + 2|+6 - 1 \)

Then combine like terms: \( 2|x + 2|+5 \)

Step4: Determine the domain of \( (f \circ g)(x) \)

The domain of a composition \( (f \circ g)(x) \) is the set of all \( x \) such that \( x \) is in the domain of \( g \) and \( g(x) \) is in the domain of \( f \).

• The domain of \( g(x)=|x + 2|+3 \): The absolute value function \( |x + 2| \) is defined for all real numbers, and adding 3 doesn't restrict the domain. So the domain of \( g \) is \( (-\infty, \infty) \).
• The domain of \( f(x)=2x - 1 \): This is a linear function, which is defined for all real numbers. So any \( g(x) \) (which is a real number for all real \( x \)) will be in the domain of \( f \).

Thus, the domain of \( (f \circ g)(x) \) is all real numbers, \( (-\infty, \infty) \).

Answer:

\( (f \circ g)(x)=2|x + 2|+5 \) and the domain of \( (f \circ g)(x) \) is \( (-\infty, \infty) \), so the correct option is the one with \( (f \circ g)(x) = 2|x + 2|+5 \) and domain \( (-\infty, \infty) \) (the third option in the given choices).