1'7 Combinations of Functions Composite Functions

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1.7 Combinations of FunctionsComposite Functions

• Objectives
• Find the domain of a function
• Combine functions using algebra.
• Form composite functions.
• Determine domains for composite functions.

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• Find the indicated function values and determine
  whether the given values are in the domain of the
  function.
• f(1) and f(5), for
• f(1)
• Since f(1) is defined, 1 is in the domain of f.
• f(5)
• Since division by 0 is not defined, the number 5
  is not in the domain of f.

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• Find the domain of the function
• Solution
• We can substitute any real number in the
  numerator, but we must avoid inputs that make the
  denominator 0.
• Solve x2 ? 3x ? 28 0.
• (x ? 7)(x 4) 0
• x ? 7 0 or x 4 0
• x 7 or x ?4
• The domain consists of the set of all real
  numbers except ?4 and 7 or xx ? ?4 and x ? 7.

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To find the domain of a function that has a
variable in the denominator, set the denominator
equal to zero and solve the equation. All
solutions to that equation are then removed from
consideration for the domain.
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Find the domain

• Since the radical is defined only for values that
  are greater than or equal to zero, solve the
  inequality

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Visualizing Domain and Range

• Keep the following in mind regarding the graph of
  a function
• Domain the set of a functions inputs, found
  on the x-axis (horizontal).
• Range the set of a functions outputs, found
  on the y-axis (vertical).

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Example

• Graph the function. Then estimate the domain and
  range.
• Domain 1, ?)
• Range 0, ?)

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The domain of a function is normally all real
numbers but there are some exceptions

• A) You can not divide by zero.
• Any values that would result in a zero
  denominator are NOT allowed, therefore the domain
  of the function (possible x values) would be
  limited.
• B) You can not take the square root (or any even
  root) of a negative number.
• Any values that would result in negatives under
  an even radical (such as square roots) result in
  a domain restriction.

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Example

• Find the domain
• There are xs under an even radical AND xs in
  the denominator, so we must consider both of
  these as possible limitations to our domain.

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Example

• Given that f(x) x 2 and g(x) 2x 5, find
  each of the following.
• a) (f g)(x) b) (f g)(5)
• Solution
• a)

b) We can find (f g)(5) provided 5 is in the
domain of each function. This is true. f(5) 5
2 7 g(5) 2(5) 5 15
(f g)(5) f(5) g(5) 7 15 22 or
(f g)(5) 3(5) 7 22
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Example

• Given that f(x) x 2 and g(x) 2x 5, find
  each of the following.
• a) (f - g)(x) b) (f - g)(5)
• Solution
• a)

b) We can find (f - g)(5) provided 5 is in the
domain of each function. This is true. f(5) 5
2 7 g(5) 2(5) 5 15
(f - g)(5) f(5) - g(5) 7 - 15 -8 or
(f - g)(5) -(5) - 3 -8
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Example

• Given that f(x) x 2 and g(x) 2x 5, find
  each of the following.
• a) (f g)(x) b) (f g)(5)
• Solution
• a)

b) We can find (f g)(5) provided 5 is in the
domain of each function. This is true. f(5) 5
2 7 g(5) 2(5) 5 15
(f g)(5) f(5)g(5) 7 (15) 105 or
(f g)(5) 2(25) 9(5) 10 105
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Given the functions below, find and give the
domain.
The radicand x 3 cannot be negative. Solving
gives
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Composition of functions

• Composition of functions means the output from
  the inner function becomes the input of the outer
  function.
• f(g(3)) means you evaluate function g at x3,
  then plug that value into function f in place of
  the x.
• Notation for composition

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Given two functions
f
and
g
, the
composite function
, denoted by
(read as
f
composed with
g
), is defined by
The domain of
is the set of
x
all numbers
in the domain of
g
(
g
such that
x
) is in the domain
of
f
.
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Suppose
and
. Find
.
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. Find
Suppose
and
the domain of
.
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Suppose that andfind
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Suppose that andfind