A relation between two sets, typically $X$ and $Y$, is a set of ordered pairs. A function is a special type of relation where every $x$-value in $X$ matches to exactly one $y$-value in $Y$. Different $x$-values can map to the same $y$-value, but each $x$-value can only be paired with one $y$-value.

In this case, the $x$ variable is your independent variable and $y$ your dependent, but different situations will see that change. The area of a circle is a function of its radius $(A=\pi r^2)$, so $A$ is the dependent variable that is determined by the independent one, $r$.

You’ll see the words ‘implicit’ and ‘explicit’ throughout the course when talking about functions. Explicit functions are what you are used to seeing. They are the ones that are evaluated for $y$.

\[y = \frac{1}{2}\left(1-x^2\right)\]And you can take that and use function notation to rewrite it.

\[f(x) = \frac{1}{2}\left(1-x^2\right)\]This is handy since it’s clear that $x$ is the independent variable, with $f$ being the function. Just keep in mind that $f(x)$ is just the popular name used. You will run to others, like our area function from before.

\[A(r) = \pi r^2\]Implicit functions are when the function is not evaluated for the dependent variable.

\[x^2 + 2y = 1\]When evaluating functions, you only need to substitute what’s in the independent variable.

\[\begin{align*} f(x) &= x^2 + 7 \\ f(3a) &= (3a)^2 + 7 \\ &= 9a^2 + 7 \end{align*}\]Finding the domain and range of a function, range of $x$ and $y$ values covered by the function, requires knowledge about the function itself. We’re not going to cover each parent function here, so I suggest using a graphing utility like Desmos to explore these functions and see what they do.

Remember that the graph of a function contains all the points $(x,f(x))$, and because they are functions, each $x$ maps to exactly one $f(x)$. The **vertical line test** can be used to determine if a graph is a function. The test fails when it crosses the graph more than once.

The book also has a collection of parents graphs on page 26 that is worth copying down.

The teal box on page 27 covers the basics in a nice neat package, but here’s the basic idea:

- adding a value to a function shifts it
- adding after the function shifts it vertically $c$ units: $f(x)+c$
- adding it before the function shifts in horizontally, but opposite of the direction you would think
- $f(x+2)$ shifts it
*left*2 units - $f(x-2)$ shifts it
*right*2 units

- $f(x+2)$ shifts it
- Negating after the function flips its vertically: $-f(x)$
- Negating before the function flips it horizontally: $f(-x)$

The book breaks down the form of a polynomial function and is worth reading in detail to re-familiarize yourself. They will come up often.

\[f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_2x^2 + a_1x + a_0\]Worth pointing is that the leading term dictates the number of potential zeros, along with the end behavior of the graph. Polynomials with an even degree will end in the same directions, while odd ones will do opposites.

Functions can be combined to form new ones. Combine them as you would any other pair of numbers, by adding them, multiplying them, and so on. The oddball out is function composition, where you run one function through another. Let’s look at $f(x) = 2x-3$ and $g(x)=\cos x$.

\[\begin{align*} (f \circ g)(x) &= 2(\cos x) - 3\\ (g \circ f)(x) &= \cos (2x-3) \end{align*}\]Functions themselves can be even or odd, which is where they are symmetrical across the $y$- and $x$-axis, respectively. Even functions have the property of $f(-x)=f(x)$, while odd functions are when $f(-x)=-f(x)$.

\[\begin{align*} f(x) &= x^3 - x \\ f(-x) &= (-x)^3 + x \\ f(-x) &= -(x^3 -x) \\ f(-x) &= -f(x) \end{align*}\]