Question
Write the function as a composition of two functions. Find two functions \( f(x) \) and \( g(x) \) such that \( y=f[g(x)] \). \( y=\left(x^{\frac{1}{2}}-4\right)^{6}+\left(x^{\frac{1}{2}}-4\right)+24 \)
Ask by Chang Montgomery. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y = f[g(x)] \), where \( g(x) = x^{\frac{1}{2}} - 4 \) and \( f(u) = u^{6} + u + 24 \).
Solution
To express the function \( y = \left(x^{\frac{1}{2}} - 4\right)^{6} + \left(x^{\frac{1}{2}} - 4\right) + 24 \) as a composition of two functions \( f(x) \) and \( g(x) \), we can follow these steps:
1. **Identify the inner function \( g(x) \)**:
- The expression \( x^{\frac{1}{2}} - 4 \) can be treated as the inner function. Thus, we can define:
\[
g(x) = x^{\frac{1}{2}} - 4
\]
2. **Identify the outer function \( f(x) \)**:
- The outer function will take the output of \( g(x) \) and apply the remaining operations. The expression becomes:
\[
f(u) = u^{6} + u + 24
\]
where \( u = g(x) \).
3. **Combine the functions**:
- Therefore, we can express \( y \) as:
\[
y = f[g(x)]
\]
In summary, the two functions are:
- \( g(x) = x^{\frac{1}{2}} - 4 \)
- \( f(u) = u^{6} + u + 24 \)
Thus, the composition is:
\[
y = f[g(x)]
\]
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Simplify this solution Extra Insights
To express the given function as a composition of two functions, we want to identify a function \( g(x) \) that simplifies our expression, and then a function \( f(u) \) that builds it back up. Let's define: - \( g(x) = x^{\frac{1}{2}} - 4 \) - \( f(u) = u^6 + u + 24 \) Now we see that if we substitute \( g(x) \) into \( f(u) \), we have: \[ y = f[g(x)] = f(x^{\frac{1}{2}} - 4) = (x^{\frac{1}{2}} - 4)^6 + (x^{\frac{1}{2}} - 4) + 24 \] Thus, the composition \( y = f[g(x)] \) is defined as desired.
