Composition of Functions

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Composition of Functions

• Lesson 4.8

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Sometimes it is necessary to write or use two or
more functions so you can answer a question or
analyze a problem.
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• Suppose an offshore oil well is leaking.
• Graph A shows the radius, r, of the spreading oil
  slick, growing as a function of time, t, so
  rf(t).
• Graph B shows the area, a, of the circular oil
  slick as a function of its radius, r, so Ag(r).

Time is measured in hours, the radius is measured
in kilometers
area is measured in square kilometers
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• Suppose you want to find the area of the oil
  slick after 4 hours.
• You can use function f on Graph A to find that
  when t equals 4, r equals 1.5.
• Next, using function g on Graph B, you find that
  when r equals 1.5, a is approximately 7.
• So, after 4 h, the radius of the oil slick is 1.5
  km and its area is 7 km2.

Time is measured in hours, the radius is measured
in kilometers
area is measured in square kilometers
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You used the graphs of two different functions, f
and g, to find that after 4 h, the oil slick has
area 7 km2. You actually used the output from
one function, f, as the input in the other
function, g. This is an example of a composition
of functions to form a new functional
relationship between area and time, that is,
ag(f(t)). The symbol g(f(t)), read g of f of
t, is a composition of the two functions f and
g. The composition g(f(t)) gives the final
outcome when an x-value is substituted into the
inner function, f, and its output value, f (t),
is then substituted as the input into the outer
function, g.
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Example A
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Function f is the inner function, and function g
is the outer function. Use equations and tables
to identify the output of f and use it as the
input of g.
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Use ideas about transformations to get the
specific equation for yg(f(x)). Use the parent
function y x, translate the vertex right 4
units, and then dilate horizontally by a factor
of 4 and vertically by a factor of 3. This gives
the equation
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You can algebraically manipulate this equation to
get the equivalent equation which is
the equation of f substituted for the input of g.
You can always create equations of composed
functions by substituting one equation into
another.
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Procedural Note

• 1. Place the mirror flat on the floor 0.5 m from
  a wall.
• 2. Use tape to attach tape measures or meter
  sticks up the wall to a height of 1.5 to 2 m.

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Looking Up

• First, youll establish a relationship between
  your distance from a mirror and what you can see
  in it.
• Set up the experiment as in the Procedure Note.
• Stand a short distance from the mirror, and look
  down into it. Move slightly left or right until
  you can see the tape measure on the wall
  reflected in the mirror.

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• Have a group member slide his or her finger up
  the wall to help locate the highest height mark
  that is reflected in the mirror. Record the
  height in centimeters, h, and the distance from
  your toe to the center of the mirror in
  centimeters, d.

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• Change your distance from the mirror and repeat
  the last step. Make sure you keep your head in
  the same position. Collect several pairs of data
  in the form (d, h). Include some distances from
  the mirror that are small and some that are
  large.
• Find a function that fits your data by
  transforming the parent function h1/d. Call
  this function f.

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• Now youll combine your work from previous steps
  with the scenario of a timed walk toward and away
  from the mirror.
• Suppose this table gives your position at
  1-second intervals
• Use one of the families of functions from this
  chapter to fit these data. Call this function g.
  It should give the distance from the mirror for
  seconds 0 to 7.

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• Let (4.7,30) represent the vertex and correspond
  vertex (0,0) on the parent function dg(t). The
  vertex has shifted 4.7 units to the right and 30
  units up.
• Let (1,112) represent another point on the
  parabola and correspond with (-1,1) on the parent
  function.
• Considering the data points (4.7, 30) and (1,
  112) that the graph as been stretched
  horizontally by a factor of 3.7 units and
  vertically by 82 units.

4.7
30
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• Combining these scale factors with the
  translations yields

4.7
30
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• Use your two functions to answer these questions
• How high up the wall can you see when you are 47
  cm from the mirror?
• Where are you at 1.3 seconds?
• How high up the wall can you see at 3.4 seconds?

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• Change each expression into words relating to the
  context of this investigation and find an answer.
  Show the steps you needed to evaluate each
  expression.
• f(60)
• g(5.1)
• f (g(2.8))

how high up the wall you can see when you are 60
cm from the mirror 123 cm
your distance from the mirror at 5.1 s 31 cm
how high you can see up the wall at 2.8 s 143 cm
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• Find a single function, H(t), that does the work
  of f(g(t)).
• Show that H(2.8) gives the same answer as Step 7c
  above.

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• To find the domain and range of a composite
  function, you must look closely at
• the domain and range of the original functions.

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Example A

• Let f (x) and g (x) be the functions graphed
  below. What is the domain of f(g(x))?

Identifying the domain of the inner function,
g(x). Domain 1x5. Range 1 g(x)3.
Identifying the domain of the outer function,
f(x). Domain -1x2. Range 0 f(x) 3.8.
These range values of g become the input for the
outer function, f(x). Notice that not all of
these output values lie in the domain of f (x),
those greater than 2.
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Example A

• Let f (x) and g (x) be the functions graphed
  below. What is the domain of f(g(x))?

Identifying the domain of the inner function,
g(x). Domain 1x5. Range 1 g(x)3.
Identifying the domain of the outer function,
f(x). Domain -1x2. Range 0 f(x) 3.8.
Now identify the x-values that produced this part
of the range of g(x) so that 1g(x)2. This is
the domain of the composite function. The domain
is 1x3.
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Example A