Mccarthy Joseph
04/09/2024 · High School
\( fog=f(x)=\sqrt {x^{2}+5}=\sqrt {x^{2}-5} \)
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Step-by-step Solution
To solve the study problem, we need to find the composition of functions \( f \circ g \), where \( f(x) = \sqrt{x^2 + 5} \) and \( g(x) = \sqrt{x^2 - 5} \).
The composition \( f \circ g \) is defined as \( (f \circ g)(x) = f(g(x)) \). So, we need to substitute \( g(x) \) into \( f(x) \).
Let's calculate \( f(g(x)) \):
1. Start with the function \( f(x) \):
\( f(x) = \sqrt{x^2 + 5} \)
2. Substitute \( g(x) \) into \( f(x) \):
\( f(g(x)) = \sqrt{g(x)^2 + 5} \)
3. Replace \( g(x) \) with \( \sqrt{x^2 - 5} \):
\( f(g(x)) = \sqrt{(\sqrt{x^2 - 5})^2 + 5} \)
4. Simplify the expression inside the square root:
\( f(g(x)) = \sqrt{x^2 - 5 + 5} \)
5. Combine like terms:
\( f(g(x)) = \sqrt{x^2} \)
6. Simplify the square root of \( x^2 \):
\( f(g(x)) = |x| \)
So, the composition \( f \circ g \) is \( |x| \). This means that when you apply the function \( f \) to the output of the function \( g \), you get the absolute value of \( x \).
Quick Answer
The composition \( f \circ g \) is \( |x| \).
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