2.1: Functions

Goals:

Describe what a function is.
Learn how to mathematically represent a function and manipulate them while avoiding common mistakes.

Functions All Around Us

The word function describes a dependence of one quantity on another. For example:

The current room temperature is a function of time.
After a flight takes off, the total distance a plane travels is a function of time since takeoff.

Functions should be thought of as a machine.

For example, when you input time, you get out the current temperature. Any temperature app you have uses this idea.

Notice that at a single time point, only one output is returned. This leads us to the mathematical definition of a function.

A function $f$ is a rule that assigns to each element $x$ in a set $A$ to exactly one element, called $f(x)$ in a set $B$.

What are some examples of functions you can think of?

Evaluating a Function

A simple way to mathematically represent input-output is with the notation $f(x)$:

everything inside the parentheses must be treated as the input.

A function is defined by the formula \[f(x) = x^2 - 2\] Evaluate the following:

$f(0)$
$f(-1)$
$f(-a)$
$f(x + h)$
Hint Everything inside the parentheses is "$x + h$".
$f(x + h) - f(x)$

From now on, when given a directive, like "evaluate $f(x + h)$" you must fully simplify.

The word "simplify" was was defined in Section 1.4.

A function is defined by \[g(x) = x^2 - x\] Evaluate the following:

$g(0)$
$g(-1)$
$g(-a)$
$g(x + h)$
$g(x + h) - g(x)$

Domain of a Function

domain
The domain of a function is the set of all possible inputs, when evaluated, gives you a real number.

Suppose $f(x) = \sqrt{x}$. Show that $x = 0$ is in the domain and $x = -1$ is not.

When finding domain, follow two steps:

Look for problems (these are numbers, when evaluated, result in something that is not a real number). Problems can be found by

Setting "denominators" $= 0$ and solving.
Setting "inside square roots" $< 0$ and solving.

Remove the problems from $\mathbb{R}$ and write your answer in interval notation.

What is the domain of $g(x) = \sqrt{x}$?

What is the domain of $g(x) = \dfrac{1}{x^2-4x}$?

What is the domain of $g(x) = \dfrac{x}{\sqrt{x + 1}}$?

Piecewise Functions

Piecewise functions are functions where you have multiple different functions on different parts of the domain.

For example, \[f(x) = \begin{cases} x^2 & x \leq 1 \\ 2x + 1 & x > 1\end{cases}\] is a piecewise function.

Here's how to read this: If your input $x$ is less than or equal to 1, then you need to plug it into $x^2$.

If your input $x$ is strictly greater than 1, plug it into $2x + 1$.

For the above function, evaluate $f(0), f(1)$ and $f(2)$.

Suppose \[f(x) = \begin{cases} 1 & x \geq 0 \\ 0 & x < 0 \end{cases}\] Evaluate $f(-2), f(-1), f(-0.01), f(0), f\left(\dfrac{1}{100}\right), f(1)$ and $f(2)$.