Question
Express the function \( G \) in the form \( f \circ g \). (There is more than one correct answer. The function is of the form \( y=f(g(x)) \). Use non-identity functions for \( f(x) \) and \( g(x) \). ) \( G(x)=\frac{1}{x+3} \) \( (f(x), g(x))=(\square \)
Ask by Todd Gonzalez. in the United States
Mar 01,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f(x) = \frac{1}{x} \) and \( g(x) = x + 3 \).
Solution
One acceptable way to write G(x) = 1/(x + 3) as a composition of two non‐identity functions is as follows:
Let g(x) = x + 3 and f(x) = 1/x.
Then, f(g(x)) = f(x + 3) = 1/(x + 3), which equals G(x).
Note that there are many possible pairs (f, g) that work; the above is just one valid example.
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Simplify this solution Beyond the Answer
One way to express the function \( G(x) = \frac{1}{x+3} \) in the form \( f(g(x)) \) is to choose \( g(x) = x + 3 \) and \( f(x) = \frac{1}{x} \). This gives us \( G(x) = f(g(x)) = f(x + 3) = \frac{1}{x + 3} \). You can also mix it up with \( g(x) = x + 3 \) and \( f(x) = \frac{1}{x} \) again, but instead choose \( g(x) = 3 + x \) and \( f(x) = \frac{1}{x} \). Both choices lead to the same original function \( G \)!
