twelve basic functions key - ms. stevenson€¦ · • neither odd or even • bounded • no local...
TRANSCRIPT
THE TWELVE BASIC FUNCTIONS KEY FUNCTION: f (x) = x SKETCH
• DOMAIN: (−∞,∞) ; all reals
• RANGE: [0,∞) • Continuous • Decreasing on: (−∞,0] • Increasing On: [0,∞) • Symmetric with respect to y-axis • Even function • Bounded Below • Local minimum at (0,0) • No Horizontal Asymptotes • No Vertical Asymptotes
• End Behavior: lim f (x)x→−∞
= ∞ ; lim f (x)x→∞
= ∞
FUNCTION: f (x) = x SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: (−∞,∞) ; all reals
• Continuous • Not Decreasing on any Interval • Increasing On: (−∞,∞) • Symmetric with respect to The Origin • Odd function • Unbounded • No local minimum or maximum since it is
increasing on the entire domain • No Horizontal Asymptotes • No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= −∞ ; lim f (x)x→∞
= ∞
FUNCTION: f (x) = x2 SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: [0,∞) • Continuous • Decreasing on: (−∞,0] • Increasing On: [0,∞) • Symmetric with respect to The y-axis • Even function • Bounded below • Local Minimum at (0,0) • No Horizontal Asymptotes • No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= ∞ ; lim f (x)x→∞
= ∞
THE TWELVE BASIC FUNCTIONS KEY FUNCTION: f (x) = x3 SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: (−∞,∞) ; all reals
• Continuous • Not Decreasing on any Interval • Increasing On: (−∞,∞) • Symmetric with respect to The Origin • Odd function • Unbounded • No local minimum or maximum since it is
increasing on the entire domain • No Horizontal Asymptotes • No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= −∞ ; lim f (x)x→∞
= ∞
FUNCTION: f (x) = 1x
SKETCH
• DOMAIN: (−∞,0)∪ (0,∞) • RANGE: (−∞,0)∪ (0,∞) • Discontinuous; However, it is
Continuous for every point in its domain • Decreasing on: (−∞,0)∪ (0,∞) • Not increasing on any interval • Symmetric with respect to The Origin • Odd function • Unbounded • No local minimum or maximum since it is
Decreasing on the entire domain • Horizontal Asymptotes: y = 0
• Vertical Asymptotes: x = 0
End Behavior: lim f (x)x→−∞
= 0 ; lim f (x)x→∞
= 0
FUNCTION: f (x) = x SKETCH
• DOMAIN: [0,∞) • RANGE: [0,∞) • Continuous • Not Decreasing on any interval • Increasing on: [0,∞) • Not symmetric • Neither Odd or Even • Bounded Below • Absolute minimum at (0,0) • No Horizontal Asymptotes • No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= D.N .E. ; lim f (x)x→∞
= ∞
THE TWELVE BASIC FUNCTIONS KEY FUNCTION: f (x) = ex SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: (0,∞) • Continuous • Not Decreasing on any interval • Increasing on: (−∞,∞); all reals
• Not symmetric • Neither Odd or Even • Bounded Below • No local minimum or maximum since it is
Increasing on the entire domain • Horizontal Asymptotes: y=0 • No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= 0 ; lim f (x)x→∞
= ∞
FUNCTION: f (x) = ln(x) SKETCH
• DOMAIN: (0,∞) • RANGE: (−∞,∞) ; all reals
• Continuous • Not Decreasing on any interval • Increasing on: (0,∞) • Not symmetric • Neither Odd or Even • Unbounded • No local minimum or maximum since it is
Increasing on the entire domain • No Horizontal Asymptotes • Vertical Asymptotes: x = 0
End Behavior: lim f (x)x→−∞
= D.N .E. ; lim f (x)x→∞
= ∞
FUNCTION: f (x) = sin(x) SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: [−1,1] • Continuous
• Decreasing on: π2, 3π2
⎡⎣⎢
⎤⎦⎥repeat 2π cycles
• Increasing on: − π2,π2
⎡⎣⎢
⎤⎦⎥repeat 2π cycles
• Symmetric to Origin • Odd Function • Bounded • Absolute Max at x = π
2+ 2πn
• Absolute Min at x = 3π2
+ 2πn
• No Horizontal Asymptotes • No Vertical Asymptotes • End Behavior: lim f (x)
x→−∞= −1 to1 ; lim f (x)
x→∞= −1 to1
THE TWELVE BASIC FUNCTIONS KEY FUNCTION: f (x) = cos(x) SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: [−1,1] • Continuous • Decreasing on: π + 2πn,2π + 2πn[ ]
• Increasing on: 2πn,π + 2πn[ ]
• Symmetric to y-axis • Even Function • Bounded • Absolute Max at x = 0 + 2πn • Absolute Min at x = π + 2πn
• No Horizontal Asymptotes • No Vertical Asymptotes End Behavior: lim f (x)
x→−∞= −1 to1 ; lim f (x)
x→∞= −1 to1
FUNCTION: f (x) = int(x) = x⎢⎣ ⎥⎦ SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: {y | y∈Z}
• Discontinuous • Not decreasing on any interval • Increasing on: (−∞,∞); all reals
• Constant on: [x, x +1) • Not symmetric • Neither Odd or Even • Unbounded • No local minimum or maximum since it is
Increasing on the entire domain • No Horizontal Asymptotes • No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= −∞ ; lim f (x)x→∞
= ∞
FUNCTION: f (x) = 11+ e−x
SKETCH
• DOMAIN: (−∞,∞); all reals
• RANGE: (0,1) • Continuous • Not decreasing on any interval • Increasing on: (−∞,∞); all reals
• Not symmetric • Neither Odd or Even • Bounded • No local minimum or maximum since it is
Increasing on the entire domain • Horizontal Asymptotes: y = 0 and y = 1
• No Vertical Asymptotes
End Behavior: lim f (x)x→−∞
= 0 ; lim f (x)x→∞
= 1