Knowledge MountainA PLACE FOR EVERY CURIOUS MINDUNIT 01 / FUNCTIONS & EQUATIONS
The Function
Workbench.
Don’t just watch a solution. Change a graph, catch a mistake, and make the next step yourself.
Algebra 2 · Unit 1.
A mountain of ideas.
27 reading, memory, and recall notes. Repeated cards build the pile; each opens its learning note.
Click a card to unfold it. Drag it onto the desk, then flick and release. Keyboard: Enter opens a card; on a loose note, R returns it. Use Pan view to move across a narrow screen.
Your notes and learning progress stay safe when the pile moves.
Inputs, outputs & the edges
A function accepts inputs from its domain and produces outputs in its range. A graph is a picture of those permitted pairs—not permission to use every number.
01Domain: read the x-values covered by the graph. Range: read the y-values it reaches. A vertical line can meet a function graph at most once.
02Use [ ] when an endpoint is included and ( ) when it is excluded. Infinity is never an endpoint you can include: always use a parenthesis beside ∞.
03For an unrestricted polynomial, the domain is all real numbers. For a nonconstant polynomial of even degree, both tails agree; for odd degree, they disagree. The leading coefficient chooses up or down.
04For an exponential bˣ with b > 0 and b ≠ 1, the domain is all real numbers and the range is (0, ∞). It approaches zero on one side but never reaches it.
05Context can shrink a domain. Counts usually use nonnegative integers; elapsed time generally uses nonnegative real numbers. Restrictions can also change the range.
Find the domain and range of f(x) = (x − 2)² + 1 for x ≥ 4.
- 1The restriction gives the domain immediately: [4, ∞).
Would you let this pass?
“The range is [1, ∞) because the vertex is at height 1.”
Say what went wrong before opening the repair.