Free public beta until December 4, 2026:
every course is open, no card needed.
After that, the first chapter of each course stays free and full access is a one-time $39 ($19 for up to 100 founding members chosen for helpful feedback). Your progress is kept.
How the beta works
Chapter 0 wrap-up
Readiness review
Closing thought
None of this chapter's tools is new, and that is the point. Functions, graphs, inverses, logs, radians, factoring and difference quotients all come from College Algebra. What changes is the speed you need with them. Calculus leans on each move in the middle of a harder thought, so each one has to cost you seconds, not minutes.
You can now look at \sqrt{3x^2+1} and name the inner piece, 3x^2+1, and the outer square root. You can also build a difference quotient and clean it up. For f(x)=x^2, the quotient \frac{(x+h)^2-x^2}{h} expands to \frac{2xh+h^2}{h}. That simplifies to 2x+h once h\ne0. Chapter 1 starts from that last expression. It asks what 2x+h heads toward as h shrinks toward zero, and turns that question into a precise tool.
Chapter review
One practice set per topic this chapter taught — same fixed, ungraded
ladders as the practice panels inside each section.
Practice: Chapter review: functions, algebra, and the launchpad
(5 problems · doesn’t affect your progress)
What is the domain of f(x)=\dfrac{1}{x^2-4}?
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
x^2-4=0 factors as (x-2)(x+2)=0, giving x=2 and x=-2, not x=4.
x=0 is fine here: 0^2-4=-4\neq0.
The denominator does hit 0 at two specific values.
Why
x^2-4=(x-2)(x+2), which is 0 at x=2 and x=-2. Every other real number is allowed.
Set the denominator x^2-4 equal to 0 and solve.
All real numbers except x=2 and x=-2
As a chapter-review refresher, evaluate the logarithm \log_2(32) by asking what power of 2 gives that result.
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
2^5=32, so \log_2(32)=5.
Correct answer
Also accepted:
Why
Ask: 2 to what power gives 32?
5
As a chapter-review check, convert 270° to radians using the standard degree-to-radian conversion factor.
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Why
g(2)=2^2=4, then f(4)=4+1=5.
Correct answer
Also accepted:
Why
Compute g(2) first, then feed that result into f.
5
As a chapter-review check, simplify the difference quotient \dfrac{f(x+h)-f(x)}{h} for f(x)=x^2+1.
Correct! 🎉Not quite.
Right on try — credit.Right — the answer had already been shown, so no credit this time.Next try is worth Answer shown after three tries — this one no longer earns credit.You chose to see the answer — this one earns no credit.
Any equivalent form counts — 2x+3, 3 + 2·x, … Use the keypad below the field for fractions, roots, and other symbols.
Why
f(x+h)-f(x)=2xh+h^2=h(2x+h), so dividing by h leaves 2x+h.
Correct answer
Also accepted:
Why
Expand f(x+h), subtract f(x), then factor out h from what's left.