A power function is a function of the form \( f(x) = ax^n \), where \( a \) and \( n \) are constants. The domain and range of a power function depend on the exponent \( n \) and the coefficient \( a \). Let's analyze the domain and range for different scenarios of \( n \).

### Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

1. **For \( n \) a positive integer**: The domain is all real numbers (\( \mathbb{R} \)).
   - Example: \( f(x) = x^2 \) has a domain of \( (-\infty, \infty) \).

2. **For \( n \) a negative integer**: The domain is all real numbers except zero, because \( x^0 \) is undefined.
   - Example: \( f(x) = x^{-2} = \frac{1}{x^2} \) has a domain of \( (-\infty, 0) \cup (0, \infty) \).

3. **For \( n \) a positive fraction (rational number)**: The domain is all non-negative real numbers (\( [0, \infty) \)), as fractional exponents represent roots, which are only real for non-negative inputs.
   - Example: \( f(x) = x^{1/2} = \sqrt{x} \) has a domain of \( [0, \infty) \).

4. **For \( n \) a negative fraction (rational number)**: The domain is all positive real numbers, as negative fractional exponents also involve roots.
   - Example: \( f(x) = x^{-1/2} = \frac{1}{\sqrt{x}} \) has a domain of \( (0, \infty) \).

### Range

The range of a function is the set of all possible output values (y-values) the function can produce.

1. **For \( n \) a positive integer**:
   - If \( n \) is even, the range is \( [0, \infty) \).
     - Example: \( f(x) = x^2 \) has a range of \( [0, \infty) \).
   - If \( n \) is odd, the range is \( (-\infty, \infty) \).
     - Example: \( f(x) = x^3 \) has a range of \( (-\infty, \infty) \).

2. **For \( n \) a negative integer**:
   - If \( n \) is even, the range is \( (0, \infty) \).
     - Example: \( f(x) = x^{-2} \) has a range of \( (0, \infty) \).
   - If \( n \) is odd, the range is \( (-\infty, 0) \cup (0, \infty) \).
     - Example: \( f(x) = x^{-3} \) has a range of \( (-\infty, 0) \cup (0, \infty) \).

3. **For \( n \) a positive fraction**:
   - The range is \( [0, \infty) \).
     - Example: \( f(x) = x^{1/2} \) has a range of \( [0, \infty) \).

4. **For \( n \) a negative fraction**:
   - The range is \( (0, \infty) \).
     - Example: \( f(x) = x^{-1/2} \) has a range of \( (0, \infty) \).

### Summary

- **Domain**: The domain varies with the exponent \( n \) being an integer or fraction (rational number), and whether \( n \) is positive or negative.
- **Range**: The range also varies with the nature of \( n \) (even/odd, positive/negative).

### Example Problems

1. Determine the domain and range of \( f(x) = x^3 \).
2. Determine the domain and range of \( f(x) = x^{-4} \).
3. Determine the domain and range of \( f(x) = x^{2/3} \).
4. Determine the domain and range of \( f(x) = x^{-3/2} \).
5. Determine the domain and range of \( f(x) = 2x^5 \).
6. Determine the domain and range of \( f(x) = -x^2 \).
7. Determine the domain and range of \( f(x) = 3x^{-1} \).
8. Determine the domain and range of \( f(x) = 4x^{1/4} \).

### Tip

Remember, the domain is about the input values \( x \) can take, while the range is about the output values \( y \) can be.

Do you have any questions or need further details?

Explore the domain and range of power functions with detailed examples and scenarios. Learn how the exponent and coefficient affect the domain and range of functions like f(x) = ax^n. Suitable for advanced high school students.

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