Drawing the range of a function often involves visually representing the set of all possible output values (y-values) the function can produce. While you don't "draw" the range directly on the graph as a separate object, you determine the range from the graph or the function's equation, and then you can represent that range using various methods, like interval notation or a number line. Here's a breakdown of how to determine and represent the range:
1. Understand the Range
The range of a function is the set of all possible output values (y-values) that the function can produce for all valid input values (x-values). Think of it as the "shadow" the function casts on the y-axis.
2. Determine the Range
There are several ways to determine the range, depending on how the function is presented:
-
From a Graph:
- Look for the lowest and highest y-values on the graph.
- If the graph extends infinitely in either direction, the range may include infinity.
- Consider any holes or discontinuities in the graph, as these can affect the range.
- Identify any horizontal asymptotes, as the function will approach these values but may not actually reach them, affecting the upper or lower bound of the range.
-
From an Equation:
- Consider the type of function:
- Linear Functions: Unless restricted, linear functions typically have a range of all real numbers.
- Quadratic Functions: Find the vertex of the parabola. The y-coordinate of the vertex will be the minimum or maximum value, and the range will extend from that value to positive or negative infinity depending on whether the parabola opens upward or downward.
- Square Root Functions: The output will always be greater than or equal to zero (or some shifted value, depending on the function).
- Rational Functions: Look for horizontal asymptotes and any vertical asymptotes that might create discontinuities, affecting the range.
- Substitute values: As suggested by one of the references, substituting various x-values into the function can give you an idea of the possible y-values. This is especially helpful for more complex functions.
- Consider the type of function:
3. Representing the Range
Once you've determined the range, you can represent it in several ways:
-
Interval Notation: This is a common and concise way to represent the range. For example:
[a, b]represents the range from a to b, including a and b.(a, b)represents the range from a to b, excluding a and b.[a, ∞)represents the range from a to positive infinity, including a.(-∞, b]represents the range from negative infinity to b, including b.(-∞, ∞)represents all real numbers.
-
Set Notation: This is another way to represent the range using set-builder notation. For example:
{y | y ≥ a}represents all y-values greater than or equal to a.
-
Number Line: You can shade a portion of the number line to represent the range. Use closed circles (●) to indicate that an endpoint is included and open circles (○) to indicate that an endpoint is excluded.
Example
Let's say you have the function f(x) = x2.
- Graph: The graph is a parabola opening upward with its vertex at (0, 0).
- Range: The lowest y-value is 0, and the function extends to positive infinity.
- Representation:
- Interval Notation:
[0, ∞) - Set Notation:
{y | y ≥ 0} - Number Line: A number line shaded from 0 (with a closed circle at 0) to the right towards positive infinity.
- Interval Notation:
Summary
While you don't directly "draw" the range onto a graph, understanding the range is crucial for interpreting the function's behavior. You determine the range by analyzing the graph or the function's equation and then represent it using interval notation, set notation, or a number line. The key is to identify the minimum and maximum y-values the function can achieve.
