Owen Hardy
10/09/2023 · 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 compositions \( f \) of and \( g \) o \( g \). Simplify your answers as much as possible. (Assume that your expressions are defined for all \( x \) in the domain of the composition
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Tutor-Verified Answer
Step-by-step Solution
To find the compositions \( f(g(x)) \) and \( g(f(x)) \), we will substitute one function into the other.
1. **Finding \( f(g(x)) \)**:
We start with the function \( g(x) = \frac{2}{5x} \). We will substitute \( g(x) \) into \( f(x) \).
The function \( f(x) \) is defined as:
\[
f(x) = x^2 + 4
\]
Now, substituting \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{2}{5x}\right) = \left(\frac{2}{5x}\right)^2 + 4
\]
Calculating \( \left(\frac{2}{5x}\right)^2 \):
\[
\left(\frac{2}{5x}\right)^2 = \frac{4}{25x^2}
\]
Therefore, we have:
\[
f(g(x)) = \frac{4}{25x^2} + 4
\]
To combine the terms, we can express 4 as a fraction with a common denominator:
\[
4 = \frac{100}{25}
\]
Thus, we can write:
\[
f(g(x)) = \frac{4}{25x^2} + \frac{100}{25} = \frac{4 + 100x^2}{25x^2}
\]
So, the simplified form of \( f(g(x)) \) is:
\[
f(g(x)) = \frac{4 + 100x^2}{25x^2}
\]
2. **Finding \( g(f(x)) \)**:
Now we will substitute \( f(x) \) into \( g(x) \).
We have:
\[
f(x) = x^2 + 4
\]
Now substituting \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(x^2 + 4) = \frac{2}{5(x^2 + 4)}
\]
This expression is already in its simplest form.
Thus, the final results for the compositions are:
\[
f(g(x)) = \frac{4 + 100x^2}{25x^2}
\]
\[
g(f(x)) = \frac{2}{5(x^2 + 4)}
\]
Quick Answer
\[
f(g(x)) = \frac{4 + 100x^2}{25x^2}
\]
\[
g(f(x)) = \frac{2}{5(x^2 + 4)}
\]
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