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Functions are used to model mathematical situations. Examples: A=πr 2 The area of a circle is a function of its radius C=5/9(F – 32) °C as a function of degrees °F Functions: What the f?
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Function notation y = 1 - x 2 f(x) = 1 - x 2 “y is a function of x” function notation f(x) is read “f of x”
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Evaluating functions f(x) = 1 - x 2 What is f(-1)? In other words, evaluate the function f(x) at x = -1. f(-1) = 1 – (-1) 2 = 1 – 1 = 0
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Domain: For what values of x is the function defined? Range: For what values of y is the function defined?
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Basic examples f(x)=ax 2 +bx+c {all real numbers} g(x)=1/x {x: x 0} *denominator cannot be 0 h(x)=√x {x: x ≥ 0} v(x)=ln(x) {x: x > 0}
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**** Notation time-out****** In mathematics, : means “such that” Ex: {x : x 0} means “the set of all x’s such that x 0”
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Domain of y = 2x + 3 {x: 2x + 3 0} “all x’s such that 2x+3 is greater than or equal to 0”
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Domain of y = {x: 2x + 3 0} Or {x: x -3/2} 1 2x + 3
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Domain of y = 2x + 3 All real numbers 3
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Domain of y = ln(2x + 3) { x: 2x + 3 > 0 }
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Domain of y = { x: 2x + 3 > 0 } 1 2x + 3
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Assignment A p.27/ 1,2,(3-23)odd,37
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SAT Problem of the Day! “700 on the SAT math!!! Heck yes, I am going to UCLA!”
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* A more CONCISE way to describe sets. ** Is used interchangeably with set notation to express domain and range.
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{x:…}INTERVAL x 3 x 3 x 3 x 3 x 3 (- , 3) (- , 3] (3, ) [3, ) (- , 3) & (3, )
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-2 x 3 -2 x 3 -2 x 3 -2 x 3 All reals (- , ) (- 2, 3] [- 2, 3) (- 2, 3) [- 2, 3]
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MIX AND MATCH!!!
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Assignment B Ignore the directions. Instead… Find the domain of each function and write it using a) set notation and b) interval notation. p.27-28/ 4,6,8,20,30,34
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SAT Problem of the Day!
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y = |x| Absolute value
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y = x 2 parabola
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y = x Even root
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y = x 3 Odd root
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1 y = e x Exponential growth
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1 y = e - x Exponential decay
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1 y = ln x
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y = 1/x
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y = 1/x 2
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y = x 2 - 2 y = x 2 y = x 2 + 3 + move up - move down
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y = |x| y = |x - 6| y = |x +3| + move left - move right
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y = log 2 x y = log 2 (x + 2) left 2
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y = x y = - x Flip about x
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1 y = e x y = - e x flip
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1 y = - ln x
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1 y = - e -x
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Given the graph of y = f(x), To graph y = f(x) ± a, Move the graph of y = f(x) up/down a units
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Given the graph of y = f(x), To graph y = f(x ± a), Move the graph of y = f(x) left/rt a units + is left, - is right!
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Given the graph of y = f(x), To graph y = -f(x), flip the graph of y = f(x) with respect to the x-axis.
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The graph of y = - f(x) is flipped about the x-axis. The graph of y = f(-x) is flipped about the y-axis.
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y = -x
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1 y = e - x
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1 y = ln (-x)
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Given the graph of y = f(x), To graph y = kf(x), Multiply all the y values of y = f(x) by k. Steeper if k > 1. Flatter if k < 1
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y = x y = 2 x steeper
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y = |x| y = 2|x| y = ½ |x|
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y = x 3 y = 2 x 3 double the y-values
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y = x 3 y = ½ x 3 half the y-values
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1 y = e x y = 2 e x 2
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Assignment C p. 28/ 47-55,83,93
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SAT Problem of the Day!
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y = |x| y = -|x - 4| + 3 flip right 4 up 3 y = -|x - 4| + 3
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y = |x| y = -2|x - 4| + 3 flip right 4 up 3 y = -2|x - 4| + 3 steeper
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y = x 2 y = -(x - 6) 2 + 1 flip right 6 up 1 y = -(x - 6) 2 + 1
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SAT Problem of the Day
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Given two functions y=f(x) and y=g(x). “f composed with g of x”
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Example: Finding composite functions. Given f(x)=2x+3 and g(x)=cos(x). Find…
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Example: Finding composite functions. Given f(x)=2x+3 and g(x)=cos(x).
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Try this… Find two functions f and g such that F(x)=
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Even vs. Odd ODD functions are symmetric with respect to the origin. EVEN functions are symmetric with respect to the y- axis.
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Even or Odd?
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Even and Odd Functions Even, odd, or neither test: The function y=f(x) is EVEN if f(-x) = f(x). The function y=f(x) is ODD if f(-x) = -f(x). Otherwise, it is neither even nor odd.
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Example Determine whether the function is even, odd, or neither. a) b) odd * cos is even even
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Assignment E p. 28-29/ 57 – 70 *Test on Unit P: Functions This Thursday! Review Assignment online www.geocities.com/mskadlac
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