Functions

A function is a mathematical rule that relates an input (usually denoted by $x$) to an output (denoted by $f(x)$). For each input value, there is exactly one output value.

The general form of a function is:

$$f(x)=\text{expression in terms of }x$$

The function:

$$f(x)=2x+3$$

will return the value of $2x+3$ for any value of $x$

Evaluating Functions

To evaluate a function, substitute a specific value of $x$ into the function.

$f(x)=2x+3\text{, find }f(4)$

$$f(4)=2(4)+3=8+3=11$$

Domain and Range

The domain is the set of all possible input values (values of $x$) for which the function is defined.

The range is the set of all possible output values (values of $f(x)$)

For $f(x)=\sqrt{x}$​, the domain is $x\ge 0$ because square roots are only defined for non-negative values. The range is also $f(x)\ge 0$

Functions and Roots

In mathematics, the roots of a function are the values of the variable that make the function equal to zero. In other words, if $f(x)$ is a function, the roots are the solutions to the equation:

$$f(x)=0$$

Roots are also known as x-intercepts of the function, as they represent the points where the graph of the function intersects the x-axis.