Understanding Function Notation and Evaluation

In mathematics, a function is a rule that assigns exactly one output to each valid input. Instead of writing equations with $y$, such as $y = 2x + 3$, we often use function notation, writing it as $f(x) = 2x + 3$.

Here, $f$ is the name of the function, $x$ is the input, and $f(x)$ (read as "f of x") represents the output.

Evaluating Functions

Evaluating a function simply means substituting a specific value for the input variable $x$ and calculating the result.

Example: If $f(x) = x^2 - 3x + 1$, find $f(-2)$.

To find $f(-2)$, replace every $x$ in the equation with $-2$: $f(-2) = (-2)^2 - 3(-2) + 1$ $f(-2) = 4 + 6 + 1$ $f(-2) = 11$

Reading Functions from Graphs and Tables

Functions aren't just equations; they can also be visually represented as graphs or organized in tables.

• Finding $f(a)$ from a graph: Locate $a$ on the x-axis, move vertically to the graph, and read the corresponding y-value.
• Finding $x$ when $f(x) = b$: Locate $b$ on the y-axis, move horizontally to the graph, and read the corresponding x-value(s). Keep in mind that a function can have multiple x-values that result in the same y-value.

Example: If asked to "find all $x$ where $f(x) = 3$" from a graph, you look along the horizontal line $y = 3$ and find the x-coordinates of all points where the graph intersects that line.

The Vertical Line Test

Not every drawn curve or relation is a function. Because a function must have exactly one output for every input, we can quickly check graphs using the Vertical Line Test.

• The Rule: Imagine drawing vertical lines through the graph. If any vertical line crosses the graph at more than one point, the relation is not a function. If every possible vertical line crosses the graph at most once, it is a function.