Functions

Functions

• ODD
• f(x) is odd if f(-x)=-f(x)
• turning points at x=a & x=-a
• rotational symmetry of order 2; 180° about origin
• Examples
• y=sinx
• y=tanx
• y=3x^&-2x^5+3x
• y=1/x
• EVEN
• f(x) is even if f(-x)=f(x)
• turning points at x=a & x=-a
• reflective symmetry in the y-axis
• Examples
• y=cosx
• y=x^2
• y=x^4
• y=3x^6-2x^4-5

Slide 2

• PERIODIC
• A periodic function repeats itself at regular intervals;
• y=sinx Period = 2π or 360°
• y=cosx Period = 2π/2 or 540°
• y=tanx Period = π or 180°
• CONTINUOUS
• There are no breaks or gaps in the graph

Slide 3

• MODULUS FUNCTIONS
• Sketch diagram
• For graphs under x-axis, change all signs
• Integrate each part seperately
• Add together the seperate signs
• INVERSE FUNCTIONS
• Let y=f(x)
• Make x the subject
• Replace y by x to get required answer

More Functions

• Let f(x) be monotonic
• If f(x) is increasing ⇒ f(xmin) ≤ range ≤ f(xmax)
• If f(x) is decreasing ⇒ f(xmax) ≤ range ≤ f(xmin)
• ROOTS
• A quadratic has real roots if b^2-4ac≥0

Slide 5

• GRAPH SKETCHING
• Find any turning points, max & min*
• Find asymptotes - curve tending towards value
• Check behaviour of f near asymptotes
• Check behaviour of f as x→+∞
• 

Slide 6

• Vertical asymptotes
• Let denominator equation = 0
• Factorise to get x-values
• NB: These CANNOT be touched or crossed!

Slide 7

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• Caption: Horizontal Asymptotes
• Caption: Oblique Asymptotes