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Let [tex]$p(x) = 8x - 3$[/tex] and [tex]$q(x) = \sqrt[5]{x} - 3$[/tex]. Find [tex][tex]$u(x)$[/tex][/tex] given that [tex]$q(x) = p(u(x))$[/tex].

[tex]u(x) = \square[/tex]

Sagot :

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]