1. What the Range Means
The range of a function is the set of all actual outputs the function produces after plugging in every allowed input. Unlike the codomain (which is chosen when defining the function), the range shows what the function truly generates.
In short: the range is the set of real results you get from the function.
2. How Range Differs from Codomain
This is a common point of confusion. Remember:
- Codomain: all outputs that are allowed in the definition.
- Range: all outputs that actually appear.
2.1. Relationship
\text{Range}(f) \subseteq \text{Codomain}(f)
3. Finding the Range from a Formula
When a function is given by a rule, the range is found by checking what outputs are possible for all valid inputs. Sometimes it is clear from the formula; sometimes we solve inequalities.
3.1. Example 1
\( f(x) = x^2 \)
The output is always non-negative.
\text{Range} = [0, \infty)
3.2. Example 2
\( f(x) = \frac{1}{x} \)
Values approach 0 but never become 0.
\text{Range} = \mathbb{R} \setminus \{0\}
3.3. Example 3
\( f(x) = \sqrt{x + 4} \)
The inside of the square root must be ≥ 0.
x + 4 \ge 0 \Rightarrow x \ge -4
\text{Range} = [0, \infty)
3.4. Example 4
\( f(x) = \sin x \)
Sine always stays between -1 and 1.
\text{Range} = [-1,1]
4. Finding the Range from a Set of Ordered Pairs
If a function is given as a set of ordered pairs, the range is simply the set of all second elements.
4.1. Example
f = \{(2,5),(3,6),(4,9)\}
Range = {5, 6, 9}
5. Range for Piecewise Functions
For piecewise definitions, find the output values for each part separately and combine them.
5.1. Example
f(x) = \begin{cases} x+1, & x < 0 \\ x^2, & x \ge 0 \end{cases}
For x < 0, output is less than 1. For x ≥ 0, output is non-negative.
So:
\text{Range} = (-\infty, 1) \cup [0, \infty)
6. Why Knowing the Range Helps
The range tells you what values a function can produce, which helps in sketching graphs, understanding restrictions, deciding whether the function is onto, and working with inverses.