Foundations Reference

Functions

A function is a rule that assigns each input exactly one output. Function notation lets us evaluate, combine, transform, and reverse those rules.

Fact Table

Idea	Meaning	Watch For
Function	Each input has exactly one output.	A vertical line can hit the graph only once.
Domain	Allowed inputs.	No zero denominators or even roots of negatives.
Range	Possible outputs.	Read the $y$-values from the graph or rule.
Composition	One function inside another.	Work from the inside outward.
Inverse	Undoes a function.	Swap $x$ and $y$, then solve for $y$.
Transformation	Moves or changes a graph.	Outside changes affect $y$; inside changes affect $x$.

Function Notation

$$f(a)=\text{output when }x=a$$

Composition

$$(f\circ g)(x)=f(g(x))$$

Inverse Test

$$f(f^{-1}(x))=x,\quad f^{-1}(f(x))=x$$

Transformation Form

$$y=a f(x-h)+k$$

A relation can fail to be a function, but a function can still have repeated outputs. The rule is one output per input, not one input per output.

If $f(x)=2x^2-3x+1$, find $f(-2)$.

Function MoveReplace every $x$ with the input, then simplify.

$$f(-2)=2(-2)^2-3(-2)+1$$$$f(-2)=8+6+1$$$$f(-2)=15$$

Find the domain of $f(x)=\dfrac{3}{x-5}$.

Function MoveExclude inputs that make a denominator zero.

$$x-5\ne 0$$$$x\ne 5$$$$\text{Domain: }(-\infty,5)\cup(5,\infty)$$

If $f(x)=x^2+1$ and $g(x)=3x-2$, find $(f\circ g)(x)$.

Function MovePut the entire inside function into the outside function.

$$(f\circ g)(x)=f(3x-2)$$$$(f\circ g)(x)=(3x-2)^2+1$$$$(f\circ g)(x)=9x^2-12x+5$$

Find the inverse of $f(x)=2x-7$.

Function MoveWrite $y=f(x)$, swap $x$ and $y$, then solve for $y$.

$$y=2x-7$$$$x=2y-7$$$$x+7=2y$$$$f^{-1}(x)=\frac{x+7}{2}$$

Describe $g(x)=-2f(x-3)+4$.

Function MoveInside changes move horizontally; outside changes stretch, reflect, and shift vertically.

$$x-3:\ \text{right }3$$$$-2:\ \text{reflect over x-axis and stretch by }2$$$$+4:\ \text{up }4$$