Let f(x) = (x^2 + x/x)^3/4
. Find the natural domain of f and the corresponding range of f.

Let's analyze the function \( f(x) = \left( \frac{x^2 + \frac{x}{x}}{1} \right)^{3/4} \) step by step to find its natural domain and range.

### 1. Simplify the Function
We begin by simplifying the expression inside the parentheses:

\[
f(x) = \left( \frac{x^2 + \frac{x}{x}}{1} \right)^{3/4}.
\]

Since \( \frac{x}{x} = 1 \) for all \( x \neq 0 \), the function can be rewritten as:

\[
f(x) = \left( x^2 + 1 \right)^{3/4}.
\]

### 2. Finding the Domain
The function inside the parentheses is \( x^2 + 1 \), which is defined for all real values of \( x \). There are no restrictions like square roots of negative numbers or divisions by zero. Therefore, the domain of the function is:

\[
\boxed{(-\infty, \infty)}.
\]

### 3. Finding the Range
Next, we analyze the range of \( f(x) = \left( x^2 + 1 \right)^{3/4} \).

- The expression \( x^2 + 1 \) is always greater than or equal to 1 since \( x^2 \geq 0 \).
- So, \( x^2 + 1 \geq 1 \) for all \( x \).
- Now, \( \left( x^2 + 1 \right)^{3/4} \) will also be greater than or equal to \( 1^{3/4} = 1 \).

Thus, the range of the function is:

\[
\boxed{[1, \infty)}.
\]

### Final Answer:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( [1, \infty) \)

Would you like further details or have any questions?

Here are 5 related questions:
1. How does changing the exponent in \( f(x) \) affect the domain and range?
2. What is the effect of shifting the function, such as \( f(x) = (x^2 + 2)^{3/4} \)?
3. How would the range change if the exponent were \( 1/2 \) instead of \( 3/4 \)?
4. How can you find the inverse of this function, if possible?
5. What happens to the function as \( x \to \infty \) and \( x \to -\infty \)?

**Tip:** When finding the range, always consider how the function behaves at extreme values of \( x \).

Let f(x) = (x^2 + x/x)^3/4. Find the natural domain of f and the corresponding range of f.

Explore how to find the domain and range of the function f(x) = (x^2 + 1)^(3/4). This solution involves algebraic simplification and understanding exponents. Suitable for high school students studying functions and their properties.

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