1.3 - New Functions From Old Functions

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1.3 - New Functions From Old Functions
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Translation f (x) k
Graph y1 x2 on your graphing calculator and
then graph y2 given below to determine the
movement of the graph of y2 as compared to y1.
Generalize the effect of k.
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Translation f (x h)
Graph y1 x2 on your graphing calculator and
then graph y2 given below to determine the
movement of the graph of y2 as compared to y1.
Generalize the effect of h.
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Horizontal and Vertical Translations f (x h)
k

• The value of h causes the graph of f(x) to
  translate left or right (horizontally). If h lt 0,
  the graph shifts h units left. If h gt 0, the
  graph shifts h units right.
• The value of k causes the graph of f(x) to
  translate up or down (vertically). If k gt 0, the
  graph shifts k units up. If k lt 0 then the graph
  shifts k units down.

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Examples

• Use the library of functions to sketch a graph of
  each of the following without using your graphing
  calculator.
• f(x) (x 3)2 2
• g(x) x 2 3

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x- and y-Axis Reflections

• The graph of y - f(x) is the same as graph of
  f(x) but reflected about the x-axis.
• The graph of y f(-x) is the same as graph of
  f(x) but reflected about the y-axis.

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Compression and Stretches

• The graph of yaf(x) is obtained from the graph
  of y f(x) by vertically stretching the graph if
  a gt 1 or vertically compressing the graph if 0
  lt a lt 1.

Verify this by graphing y1 x y2 3x y2
(?) x
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The Algebra of Functions
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The sum f g is the function defined by (f
g)(x) f(x) g(x) The domain of f g consists
of numbers x that are in the domain of both f and
g (the intersection of the domains).
The difference f - g is the function defined by
(f - g)(x) f(x) - g(x) The domain of f - g
consists of numbers x that are in the domain of
both f and g (the intersection of the domains).
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The product f g is the function defined by (f
g)(x) f(x) g(x) The domain of f g
consists of numbers x that are in the domain of
both f and g (the intersection of the domains).
The quotient f / g is the function The
domain of f / g consists of all x such that x is
in the domain of f and g and g(x) ? 0 (the
intersection of the domains).
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Composition of Functions
Given two functions f and g, the composite
function is defined by (f ? g)(x) f(g(x)) Read
f composite g of x The domain of f ? g is the
set of all numbers x in the domain of g such that
g(x) is in the domain of f. Note In general (f
? g)(x) ? (g ? f)(x)
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Range of g
(equal to or a subset of)
Domain of g
Domain of f
Range of f
g
f
x
g(x)
f(g(x))
Domain of f(g)
Range of f(g)
(f ? g)(x) f(g(x)) Or f(g)
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Composition of Functions
Let f(x) 2x 3 and g(x) x 2 5x. Determine
(f ? g)(x).

(f ? g)(x) f(g(x)) f(x 2 5x)
2 3
x
(x 2 5x)
2x 2 10x 3
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Examples

• (a) Let and g(x) x2
  2. Determine (i)
• (f ? g)(x), (ii) (g ? f)(x), and (iii) (g ?
  g)(x).
• If y cos (x2 2) and y (f ? g)(x), determine
  f and g.
• If y esin(x5) and y (f ? g ? h)(x),
  determine f, g, and h.