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Chapter 1: Problem 39

Find (a) \(f \circ g\) and (b) \(g \circ f .\) Find the domain of each function and each composite function. $$f(x)=|x|, \quad g(x)=x+6$$

Short Answer

Expert verified
The composite function \(f \circ g(x) = |x + 6|\). The composite function \(g \circ f(x) = |x| + 6\). The domain of \(f(x)\), \(g(x)\), \(f \circ g\) and \(g \circ f\) is all real numbers.

Step by step solution

01

Calculation of \(f \circ g\)

This is calculated by replacing all \(x\) in the function \(f(x)\) with the function \(g(x)\). So, \(f \circ g(x) = f(g(x)) = |g(x)| = |x+6|\.
02

Calculation of \(g \circ f\)

This is calculated by substituting all \(x\) in the function \(g(x)\) with the function \(f(x)\). Therefore, \(g \circ f(x) = g(f(x)) = f(x) + 6 = |x| + 6\.
03

Domain of \(f(x)\)

The domain of \(f(x)\) is all real numbers, given that absolute value of \(x\) is defined for every real number.
04

Domain of \(g(x)\)

The domain of \(g(x)\) is also all real numbers since \(x + 6\) is defined for every real number.
05

Domain of \(f \circ g\)

The domain of \(f \circ g\) is all real numbers. This is because the composition function \(|x + 6|\) is also defined for all real numbers.
06

Domain of \(g \circ f\)

The domain of \(g \circ f\) is all real numbers as well, as \(|x| + 6\) is also defined for all real numbers.

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