Domain and Range of Functions: College Algebra Study Notes

Functions

Domain and Range of a Function

Understanding the domain and range of a function is fundamental in College Algebra. The domain refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values) that the function can produce.

• Domain: The set of all real numbers x for which the function f(x) is defined.

• Range: The set of all real numbers y that the function f(x) can output.

Example: Quadratic Function

Consider the graph of a quadratic function (parabola opening upwards) as shown in the image. The vertex of the parabola is the lowest point, and the arms extend infinitely upward.

• Domain: For most quadratic functions, the domain is all real numbers, since you can substitute any real value for x.

• Range: The range is determined by the vertex. If the vertex is at y = k, and the parabola opens upwards, then the range is $[k, \infty)$.

Interval Notation:

• Domain: $(-\infty, \infty)$

• Range: $[k, \infty)$, where k is the y-coordinate of the vertex.

Steps to Find Domain and Range from a Graph

1. Identify the leftmost and rightmost points on the graph to determine the domain.

2. Identify the lowest and highest points on the graph to determine the range.

3. Express the domain and range in interval notation.

Additional info:

• For quadratic functions of the form $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$ and the minimum (or maximum) value is $f(-\frac{b}{2a})$.

• If the parabola opens downwards (a < 0), the range would be $(-\infty, k]$.