Revision sheet

Function notation

• $f(a)$ is the output for the input $a$.
• Replace every $x$ by $(a)$, with parentheses: $f(-2)$ for $x^2$ is $(-2)^2 = 4$.
• On a graph: the height above $x = a$.

Watch out: $f(a + b)$ is not $f(a) + f(b)$.

Average rate of change

• From $a$ to $b$: $\dfrac{f(b) - f(a)}{b - a}$, output change over input change.
• Increasing: the graph rises from left to right.
• Give intervals of increase with $x$-values.

Watch out: change in $f(x)$ on top, change in $x$ below.

Domain and range

• Domain: the inputs. Leave out zero denominators and negatives under a square root.
• Range: the outputs.
• $[a, \infty)$ includes $a$; $(a, \infty)$ does not.

Watch out: the range uses $y$-values, not the $x$ of a vertex.

Piecewise functions

1. Find the piece whose condition the input satisfies.
2. Use that piece's formula.
3. $|x| = x$ for $x \ge 0$, and $-x$ for $x < 0$.

Watch out: at a boundary, read $<$ and $\ge$ carefully.

Inverse functions

1. Write $y = f(x)$.
2. Solve for $x$, undoing the last operation first.
3. Rename: $f^{-1}(x)$. Check that $f(f^{-1}(x)) = x$.

Watch out: $f^{-1}(x)$ is not $\dfrac{1}{f(x)}$.

Shifts and reflections

• $f(x - h)$: $h$ units right. $f(x) + k$: $k$ units up.
• $-f(x)$: flip across the $x$-axis. $f(-x)$: flip across the $y$-axis.
• Inside $f(\dots)$ acts on $x$; outside acts on $y$.

Watch out: $f(x + 3)$ moves the graph left.

Sequences

• Explicit: $a_n$ straight from $n$.
• Recursive: a first term and a rule from $a_n$ to $a_{n+1}$.

Watch out: check your formula with $n = 1$.