Understanding Function Concepts and Domain

In mathematics, a function is like a machine: you put a number in (the input), the machine applies a specific rule, and it gives you exactly one number out (the output). The most important rule of a function is that every input must have exactly one output.

Function Notation

We usually write functions using notation like $f(x)$, which is read as "$f$ of $x$".

• $x$ represents the input.
• $f(x)$ represents the output (often thought of as the $y$-value).

Example: Evaluating a Function

Problem: If $f(x) = 3x - 2$, find $f(5)$.

To find $f(5)$, substitute the input value $5$ everywhere there is an $x$ in the rule: $f(5) = 3(5) - 2$ $f(5) = 15 - 2$ $f(5) = 13$

So, when the input is $5$, the output is $13$.

Domain and Range

Every function has a set of numbers it can work with and a set of numbers it can produce.

• Domain: The set of all valid inputs ($x$-values) for which the function is defined.
• Range: The set of all possible outputs ($y$-values) the function can produce.

Finding the Domain from an Equation

For many functions, the domain is "all real numbers." However, there are two main rules that restrict the domain:

1. Fractions: The denominator cannot be zero.
2. Square Roots: You cannot take the square root of a negative number (in the real number system).

Example: Determining Domain

Problem: Determine the domain of $f(x) = \sqrt{x - 3}$.

Because we cannot take the square root of a negative number, the expression inside the square root must be greater than or equal to zero: $x - 3 \ge 0$

Add $3$ to both sides to solve for $x$: $x \ge 3$

Answer: The domain is all real numbers where $x \ge 3$.

Identifying Domain and Range from Graphs

When looking at the graph of a function:

• To find the domain, scan the graph from left to right along the $x$-axis to see which $x$-values have a corresponding point on the line or curve.
• To find the range, scan the graph from bottom to top along the $y$-axis to see which $y$-values are covered by the graph.