QuestionJuly 26, 2025

Use the pair of functions to find f(g(x)) and g(f(x)) Simplify your answers. f(x)=x^2+2, g(x)=sqrt (x+3) f(g(x))=square g(f(x))=square

Use the pair of functions to find f(g(x)) and g(f(x)) Simplify your answers. f(x)=x^2+2, g(x)=sqrt (x+3) f(g(x))=square g(f(x))=square
Use the pair of functions to find f(g(x)) and g(f(x)) Simplify your answers. f(x)=x^2+2, g(x)=sqrt (x+3) f(g(x))=square g(f(x))=square

Solution
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Answer

f(g(x)) = x + 5 ### g(f(x)) = \sqrt{x^2 + 5} Explanation 1. Calculate f(g(x)) Substitute g(x) = \sqrt{x+3} into f(x). So, f(g(x)) = (\sqrt{x+3})^2 + 2. Simplify: (\sqrt{x+3})^2 = x+3, so f(g(x)) = x + 3 + 2 = x + 5. 2. Calculate g(f(x)) Substitute f(x) = x^2 + 2 into g(x). So, g(f(x)) = \sqrt{(x^2 + 2) + 3}. Simplify: g(f(x)) = \sqrt{x^2 + 5}.

Explanation

1. Calculate $f(g(x))$<br /> Substitute $g(x) = \sqrt{x+3}$ into $f(x)$. So, $f(g(x)) = (\sqrt{x+3})^2 + 2$.<br /> Simplify: $(\sqrt{x+3})^2 = x+3$, so $f(g(x)) = x + 3 + 2 = x + 5$.<br /><br />2. Calculate $g(f(x))$<br /> Substitute $f(x) = x^2 + 2$ into $g(x)$. So, $g(f(x)) = \sqrt{(x^2 + 2) + 3}$.<br /> Simplify: $g(f(x)) = \sqrt{x^2 + 5}$.
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