Objective: Students will review functions and compositions'

1
Objective Students will review functions and
compositions.

• Warm-up p 271 (2-4, 6-8, 10-12)

2
Composition of Functions
And
Inverse Functions
3
Evaluating a Composition of Two Functions
When two functions are applied in succession,
the resulting function is called the composite
of the two given functions.
Given f(x) 8x - 1 and g(x) 3x 4, find the
following
b) (g o f)(2)
This tells you to find the value of g(x) first,
then use this value for the function f(x).
Solve for g(2).
a) (f o g)(2)
(g o f)(x) g(f(x))
(f o g)(x) f(g(x))
f(x) 8x - 1 f(2) 8(2) - 1 15
g(x) 3x 4 g(2) 3(2) 4 10
With the value of g(2) 10, you now solve for
f(10).
(g o f)(2) g(f(2)) g(15)
(f o g)(2) f(g(2)) f(10)
g(2) 10
g(x) 3x 4 g(15) 3(15) 4 49
f(x) 8x - 1 f(10) 8(10) - 1 79
(f o g)(2) 79
(g o f)(2) 49
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Evaluating a Composition of a Function With Itself
Given h(x) 4x 3, find the following
a) (h o h)(-3)
b) (h o h)(x)
(h o h)(x) h(h(x)) h(4(x)
3) h(4x 3) h(4x 3) 4(4x
3) 3 16x 12 3
16x 15
(h o h)(-3) h(h(-3))
h(4(-3) 3) h(-9)
h(-9) 4(-9) 3 -33
5
Inverse Functions
The inverse of a function is a function which
undoes what the original function did. Its
graph is the reflection of the original graph
across the line y x. The graph can be found by
interchanging the coordinates of the ordered
pairs in the original function.
The inverse of a function is written as f -1(x)
and read as the inverse of f at x or f
inverse of x.

• When x and y are interchanged in the equation
• of a function
• The coordinates of the points that satisfy the
• equation are interchanged.
• The graph of the function is reflected in the
  line y x.

• To determine the inverse of a function
• Interchange x and y in the equation of the
  function.
• Solve the resulting equation for y.

6
Graphing the Inverse Function
Note If the ordered pair (3, 6) is on the graph
of the function f(x), then the ordered pair (6,
3) will be on the graph of the inverse function,
f -1(x).
Example Find the inverse of the function f(x)
4x - 7.
y 4x - 7
x 4y - 7 x 7 4y x 7 y 4
Interchange the x and y values.
f-1(x)
(-3, 1)
(-7, 0)
y x
f(x)
(1, -3)
(0, -7)
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Verifying an Inverse
If two functions f(x) and g(x) are inverses of
each other, then f(g(x)) must equal x
AND g(f(x)) must equal x.
Verify that the functions f(x) 4x - 7 and
are inverses.
f(g(x)) must be equal to x.
g(f(x)) must also be equal to x.
f(x) 4x - 7
g(4x - 7)
(x 7) - 7 x
x
Since f(g(x)) and g(f(x)) are both equal to x,
then f(x) and g(x) are inverses of each other.
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Graphing a Function and Its Inverse
Graph f(x) x2 1 and its inverse.
(-2, 5)
(2, 5)
The graphs are symmetrical about the line y x.
(1, 2)
For the function
(-1, 2)
(5, 2)
Domain Range
(2 , 1)
(0, 1)
y gt 1
(1, 0)
For the inverse
x gt 1
Domain Range
(2, - 1)
y x
(5, -2)
Is the inverse a function?
Could the domain of f(x) be restricted so that
the inverse is a function?
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To find the inverse of a Function

• Use the Horizontal Line Test to see if an inverse
  function exists. (That is, will the inverse be a
  function?)
• Replace f(x) with y.
• Exchange x and y then solve for y.
• Replace y with f-1(x).
• To check, calculate f(f-1(x)) and f-1(f(x)) to
  see if they both equal x.
• You can also check by graphing. gt Are the graphs
  symmetric across the line yx?

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Replace f(x) with y.
Exchange x and y.
Solve for y.
Replace y with f-1(x).
Check.
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Check by graphing.
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The graph of f(x) and the graph of f-1(x) should
reflect over the line yx.
yf(x)
yx
yf-1(x)
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Check with composition of functions.
f f-1
f-1 f
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homework

• Page 278 (1-20)