Stewart Calculus · 8th Edition
Chapter 1: Functions & Models
Every key formula from Stewart Calculus Chapter 1, in one searchable page. Click a card to study it — worked examples included.
1.1 · FUNCTION BASICS
Symmetry
Even: f(−x) = f(x) · Odd: f(−x) = −f(x)
KEY NOTES
- Test f(−x) to check even/odd — don’t trust your eyes alone.
WORKED EXAMPLE
Find the domain of f(x) = /(x − 1).
Need x + 2 ≥ 0 and x ≠ 1 ⇒ [−2, 1) ∪ (1, ∞).
Need x + 2 ≥ 0 and x ≠ 1 ⇒ [−2, 1) ∪ (1, ∞).
1.2 · ALGEBRAIC FUNCTIONS
Linear
f(x) = mx + b, m = y2 − y1x2 − x1
Power · Rational · Root
f(x) = xa · f(x) = P(x)/Q(x), Q(x) ≠ 0
WORKED EXAMPLE
Line through (2, −1) and (4, 5): m = (5 − (−1))/(4 − 2) = 3.
⇒ y + 1 = 3(x − 2), i.e. y = 3x − 7.
⇒ y + 1 = 3(x − 2), i.e. y = 3x − 7.
- Domain first: denominators ≠ 0, even roots need radicand ≥ 0, logs need argument > 0.
- Not sure it’s a function? Run the Vertical Line Test.
Trigonometric
sin x, cos x: period 2π, range [−1, 1]
sin²x + cos²x = 1 · tan x = sin xcos x, period π, asymptotes x = π/2 + nπ
Exponential
f(x) = bx (b>0, b≠1): domain ℝ, range (0, ∞), through (0, 1)
Logarithm
y = logb x ⇔ by = x; domain (0, ∞) · ln x = loge x
Laws of logarithms
log xy = log x + log y · log(x/y) = log x − log y
log xr = r log x
log xr = r log x
KEY NOTES
- eln x = x (x>0) and ln(ex) = x — they undo each other.
1.3 · TRANSFORMATIONS
Shifts
y = f(x) + c up c · y = f(x) − c down c
y = f(x − c) right c · y = f(x + c) left c
y = f(x − c) right c · y = f(x + c) left c
Stretches
y = c·f(x): vertical ×c · y = f(cx): horizontal ×1/c
Reflections
y = −f(x) over the x-axis · y = f(−x) over the y-axis
WORKED EXAMPLE
From y = x2: shift right 2, reflect over the x-axis, shift up 1.
y = x2 → y = (x − 2)2 → y = −(x − 2)2 + 1.
y = x2 → y = (x − 2)2 → y = −(x − 2)2 + 1.
1.3 · COMBINING & COMPOSITION
WORKED EXAMPLE
f(x) = √x, g(x) = x + 1:
(f∘g)(x) = √(x+1), x ≥ −1 · (g∘f)(x) = √x + 1, x ≥ 0
⇒ f∘g ≠ g∘f — order matters!
(f∘g)(x) = √(x+1), x ≥ −1 · (g∘f)(x) = √x + 1, x ≥ 0
⇒ f∘g ≠ g∘f — order matters!