To express each transformation in terms of $f(x) = 2x + 1$, $g(x) = x^5$, and $h(x) = \frac{1}{x}$, let's analyze each expression.

Solution:

1. $x \to (2x+1)^5$:

   • This expression can be written as a composition of $f(x)$ and $g(x)$.
   • Specifically, let $f(x) = 2x + 1$ and then apply $g(x) = x^5$ to this result.
   • Thus, we have:
     $x \to g(f(x)) = (2x + 1)^5$

2. $x \to x^{1/5}$:

   • Here, we want a function that gives the fifth root of $x$.
   • We can express this as the inverse of $g(x) = x^5$.
   • So, we write:
     $x \to g^{-1}(x) = x^{1/5}$

3. $x \to \frac{2}{x - 1}$:

   • This expression can be created using a combination of $h(x) = \frac{1}{x}$ and a modified form of $f(x) = 2x + 1$.
   • First, shift $x$ by subtracting 1, then invert it with $h(x)$, and finally multiply by 2.
   • We express it as:
     $x \to 2 \cdot h(x - 1) = \frac{2}{x - 1}$

Would you like further details on any of these transformations, or do you have any questions?

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Related Questions:

1. How do you find the inverse of a function like $g(x) = x^5$?
2. How can compositions of functions like $f(g(x))$ be simplified further?
3. What would the derivative of $g(f(x))$ be in terms of $x$?
4. How can we express $x \to (2x + 1)^{1/5}$ in terms of $f$ and $g$?
5. How do you determine if a function has an inverse?

Tip: When dealing with compositions, always apply the inner function first, then the outer function, as this order affects the final result.