To find [tex]\( u(x) \)[/tex] given that [tex]\( q(x) = p(u(x)) \)[/tex], we can proceed as follows:
1. Define the given functions:
- [tex]\( p(x) = 8x - 3 \)[/tex]
- [tex]\( q(x) = \sqrt[5]{x} - 3 \)[/tex]
2. Express the relationship [tex]\( q(x) = p(u(x)) \)[/tex]:
[tex]\[
\sqrt[5]{x} - 3 = 8u(x) - 3
\][/tex]
3. Isolate [tex]\( u(x) \)[/tex]:
- First, eliminate the constant term [tex]\(-3\)[/tex] from both sides:
[tex]\[
\sqrt[5]{x} - 3 + 3 = 8u(x) - 3 + 3
\][/tex]
Simplifies to:
[tex]\[
\sqrt[5]{x} = 8u(x)
\][/tex]
- Next, solve for [tex]\( u(x) \)[/tex]:
[tex]\[
u(x) = \frac{\sqrt[5]{x}}{8}
\][/tex]
Hence, the function [tex]\( u(x) \)[/tex] is:
[tex]\[
u(x) = \frac{\sqrt[5]{x}}{8}
\][/tex]
So, the answer is:
[tex]\[
u(x) = \boxed{\frac{\sqrt[5]{x}}{8}}
\][/tex]
