1. **Determine the Domain**: First, identify the domain of the function, which represents all the possible input values (x-values) for the function. The domain may be explicitly given or restricted by the nature of the function.
2. **Find the Output Values**: Use the given function to calculate the corresponding output values (y-values) for each input value in the domain.
3. **Identify the Range**: Once you've determined all the possible output values, the range of the function is the set of all these output values.
4. **Check for Restrictions**: Sometimes, there may be restrictions on the output values due to the nature of the function or domain restrictions. For example, if the function represents a square root or logarithmic function, the output values may be restricted to non-negative numbers.
Let's illustrate this process with an example:
**Example**: Find the range of the function \( f(x) = x^2 \) for all real numbers.
1. **Determine the Domain**: The domain of the function \( f(x) = x^2 \) is all real numbers, since you can square any real number.
2. **Find the Output Values**: For each input value \( x \), calculate the corresponding output value \( f(x) \) by squaring \( x \).
3. **Identify the Range**: The range of the function will be all possible output values obtained from squaring real numbers. Since squaring any real number can result in a non-negative output, the range of the function is all non-negative real numbers, or \( [0, +\infty) \).
So, for the function \( f(x) = x^2 \), the range is \( [0, +\infty) \), representing all non-negative real numbers.
Keep in mind that finding the range may involve analyzing the properties of the function and understanding its behavior for different input values. Additionally, some functions may have more complicated ranges that require further analysis or techniques to determine.