Real Functions — Piecewise, Inverse, Absolute Value

Cheat Sheet · Grade 11 Math
Emil Oliversen
PIECEWISE FUNCTIONS
Different rules for different parts of domain
  • ● closed dot = included (≤, ≥)
  • ○ open dot = excluded (<, >)
  • One y per x (still a function!)
GRAPHING STEPS
  • Identify each piece and its type
  • Table: include boundary + midpoint
  • Mark open/closed at each boundary
  • Graph each piece separately
  • NEVER connect across boundaries
CONTINUITY
Continuous
No jumps (pencil never lifts)
Discontinuous
Breaks at boundary points
INVERSE FUNCTIONS
f⁻¹ undoes f · Domain↔Range swap
Key property
f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Graphs
Reflect across y = x
Points
(x,y) on f → (y,x) on f⁻¹
HOW TO FIND INVERSE
  • Step 0: replace f(x) with y
  • Step 1: swap x and y
  • Step 2: solve for y
  • Step 3: restrict domain if needed
BY FUNCTION TYPE
OriginalInverseFunction?
LinearLinearYes
QuadraticSq root (sideways)Only if restricted
Square rootQuadratic (restricted)Yes, with restriction
Abs valueTwo linear piecesOnly if restricted
ABSOLUTE VALUE
|x| = c (c>0)
x = c OR x = −c
|x| = 0
x = 0
|x| = −c
No solution
ABS VALUE INEQUALITIES
|x| < c
−c < x < c (AND)
|x| > c
x > c OR x < −c
Flip inequality when dividing by negative!
RATIONAL FUNCTIONS
Form
f(x) = a/(b(x−h)) + k
Vert. asymptote
x = h
Horiz. asymptote
y = k
Domain
ℝ except x = h
Range
ℝ except y = k
KEY REMINDERS
  • Inverse exists as function iff f is 1-to-1
  • Horizontal line test for 1-to-1
  • Solve |3x−1|=26: split into two cases
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