Lesson 3: Function Notation

Property

We use a letter like $f$ or $g$ to name a function. The notation $f(x)$, read '$f$ of $x$', represents the output value of the function $f$ when the input is $x$. If $y$ is the output variable, we can write $y = f(x)$. The parentheses in $f(x)$ do not indicate multiplication.

Function Notation:
Input variable
↓
$f(x) = y$
↑
Output variable

Examples

• Instead of writing 'the area $A$ for a radius $r$ is $\pi r^2$', we can write $A(r) = \pi r^2$. The notation $A(3)$ asks for the area of a circle with a radius of 3.

Property

Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
To evaluate a function described by an equation, we substitute the given input value into the equation to find the corresponding output, or function value.

Examples

• Given the function $f(x) = 5x - 2$, to evaluate $f(3)$, we substitute $x=3$ to get $f(3) = 5(3) - 2 = 15 - 2 = 13$.

• For the function $g(t) = t^2 + 10$, evaluating at $t=-4$ means calculating $g(-4) = (-4)^2 + 10 = 16 + 10 = 26$.

Property

The graph of a function is the graph of all its ordered pairs, $(x, y)$ or using function notation, $(x, f(x))$ where $y = f(x)$.

Examples

• For the function $f(x) = 2x + 1$, the point $(3, 7)$ is on its graph because $f(3) = 2(3) + 1 = 7$.