Functions and Their Graphs

A function $f$ assigns to each input $x$ (from the domain) exactly one output $y=f(x)$. The set of all outputs is the range.

• Evaluating: substitute the input for every $x$. In $f(x)=x^2-3x$, we get $f(a+1)=(a+1)^2-3(a+1)$.
• Domain of a polynomial: all reals, $(-\infty,\infty)$.
• Radical $\sqrt{g(x)}$: require $g(x)\ge 0$.
• Rational $\dfrac{p(x)}{q(x)}$: require $q(x)\ne 0$.

The difference quotient of $f$ is

It measures the average rate of change of $f$ over $[x,x+h]$ and is the foundation of the derivative. Simplify until the $h$ in the denominator cancels.

• $x$-intercepts (zeros): where $f(x)=0$; $y$-intercept: the value $f(0)$.
• Even (symmetric about $y$-axis): $f(-x)=f(x)$.   Odd (symmetric about origin): $f(-x)=-f(x)$.
• Increasing / decreasing / constant over an interval as $x$ moves left to right.
• Relative maximum / minimum: a peak or valley of the graph.

Constant $f(x)=c$; Identity $f(x)=x$; Absolute value $f(x)=|x|$; Square root $f(x)=\sqrt{x}$; Quadratic $f(x)=x^2$; Cubic $f(x)=x^3$; Reciprocal $f(x)=\dfrac{1}{x}$; Greatest integer (step) $f(x)=\lfloor x\rfloor$.

Starting from $y=f(x)$:

• $+d$: shift up $d$;   $-d$: shift down.
• $f(x-c)$: shift right $c$;   $f(x+c)$: shift left. (Horizontal moves are opposite the sign.)
• $a<0$: reflect over the $x$-axis;   $|a|>1$: vertical stretch;   $0<|a|<1$: vertical shrink.
• $b<0$: reflect over the $y$-axis;   $|b|>1$: horizontal shrink;   $0<|b|<1$: horizontal stretch.

For functions $f$ and $g$:

The composition $(f\circ g)(x)=f\big(g(x)\big)$ feeds the output of $g$ into $f$.

$f$ is one-to-one if each output comes from exactly one input; equivalently it passes the horizontal line test. Only one-to-one functions have an inverse $f^{-1}$, which satisfies

The graph of $f^{-1}$ is the reflection of the graph of $f$ over the line $y=x$.

A piecewise function uses different formulas on different parts of the domain. A closed dot $\bullet$ includes an endpoint; an open dot $\circ$ excludes it. Evaluate by choosing the rule whose condition the input satisfies.

Here $f(0)=1$ and $f(1)=3$.

The average rate of change of $f$ from $x=a$ to $x=b$ is the slope of the secant line:

A functional equation constrains an unknown function $f$ by an identity that must hold for all allowed $x$. The key tactic is substitution: replace $x$ with a related expression (such as $\tfrac{1}{x}$, $-x$, or $1-x$) to generate a second equation, then solve the resulting system for $f(x)$ as if $f(x)$ and the substituted value were two separate unknowns.

• Common self-inverse swaps: $x\mapsto \tfrac{1}{x}$, $x\mapsto -x$, $x\mapsto 1-x$.
• After substituting, treat $f(x)$ and $f(\text{new input})$ as variables and eliminate.

Composing $f$ with itself $n$ times is written

(The bracket distinguishes this from a power $f(x)^n$ or the inverse $f^{-1}$.) A function is an involution if $f\circ f=\mathrm{id}$, i.e. $f\big(f(x)\big)=x$; then $f^{-1}=f$ and the iterates cycle: $f^{[\text{even}]}=\mathrm{id}$ and $f^{[\text{odd}]}=f$. Some maps have longer periods (e.g. $f(x)=1-\tfrac{1}{x}$ satisfies $f^{[3]}=\mathrm{id}$).

Three high-leverage ideas:

• Monotonicity $\Rightarrow$ invertibility. If $f$ is strictly increasing (or strictly decreasing) on its domain, it automatically passes the horizontal line test, so $f^{-1}$ exists there — no need to solve algebraically to know it is one-to-one.
• Exploiting even/odd symmetry. An even function is fully determined by its values on $[0,\infty)$; an odd function must have $f(0)=0$ (when $0$ is in the domain). Products/quotients obey: even$\cdot$even $=$ even, odd$\cdot$odd $=$ even, even$\cdot$odd $=$ odd.
• Composing transformations into one mapping. Applying transformation $T_1$ then $T_2$ means the combined rule is $T_2\circ T_1$. Order matters: a shift-then-stretch gives a different single formula $y=a\,f(bx-c)+d$ than stretch-then-shift.