Martinez Ortega
04/01/2024 · Primary School
Suppose that the functions \( f \) and \( g \) are defined as follows, \[ f(x)=x^{2}+4 \] \( g(x)=\frac{2}{5 x}, x \neq 0 \) Find the composition of a function with itself Simplify your answers as much as possible, \( g \) g, (Assume that your expressions are defined for all \( x \) in the domain of the composition. You do not have to indicate (
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Step-by-step Solution
To find the composition of the function \( g \) with itself, denoted as \( g(g(x)) \), we start with the definition of \( g(x) \):
\[
g(x) = \frac{2}{5x}
\]
Now, we need to substitute \( g(x) \) into itself:
\[
g(g(x)) = g\left(\frac{2}{5x}\right)
\]
Next, we will substitute \( \frac{2}{5x} \) into the function \( g \):
\[
g\left(\frac{2}{5x}\right) = \frac{2}{5\left(\frac{2}{5x}\right)}
\]
Now, simplify the expression inside the denominator:
\[
5\left(\frac{2}{5x}\right) = \frac{10}{5x} = \frac{2}{x}
\]
Thus, we have:
\[
g\left(\frac{2}{5x}\right) = \frac{2}{\frac{2}{x}} = 2 \cdot \frac{x}{2} = x
\]
Therefore, the composition \( g(g(x)) \) simplifies to:
\[
g(g(x)) = x
\]
In conclusion, the result of the composition \( g(g(x)) \) is:
\[
\boxed{x}
\]
Quick Answer
\( g(g(x)) = x \)
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