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 \(x \geq 0\) because square roots are only defined for non-negative values. The range is also \(f(x) \geq 0\)

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.