Algebra II Chapter 6 Notes and Practice -Packet 3- Evaluation 2
Arithmetic Combinations of Functions
Recall that a function is a two variable relationship. One variable is referred to as the
input variable (domain) and the other is referred to as the output variable (range) Each input
variable can only be paired with one output variable.
Function notation: f ( x)  3 x  1
or
g ( x)  4 x 2  2
Function Operations
Functions can be put together to form new functions by adding, subtracting, multiplying,
or dividing two different functions.
( f  g )( x)  f ( x)  g ( x)
( f  g )( x)  f ( x)  g ( x)
( f  g )( x)  f ( x)  g ( x)
 f 
f ( x)
  ( x) 
g ( x)
g
g ( x)  0
Examples
If
f ( x)  3 x  1 or
g ( x)  4 x 2  2 find the following:
( f  g )( x ) = f ( x)  g ( x) = 3 x  1  4 x 2  2 = 4 x 2  3x  1
( f  g )( x)  f ( x)  g ( x)  3x  1  (4 x 2  2)  3x  1  4 x 2  2  4 x 2  3x  3
( f  g )( x)  f ( x)  g ( x)  (3x  1)(4 x 2  2)  12 x3  6 x  4 x 2  2  12 x3  4 x 2  6 x  2
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Composite Functions
f ( x)  3 x  1 or
g ( x)  4 x 2  2
Another type of function that can be formed is when the output of one function is made the input
of a second function. This is called a composite function. This function notation is written:
( f g )( x)  f ( g ( x)) This notation means to replace the input of the “f” function with the output
of the “g” function. Using the “f” and “g” function from above:
f ( g ( x))  3(4 x 2  2)  1  12 x 2  6  1  12 x 2  5
g ( f ( x))  4(3x  1)2  2  4(9 x 2  6 x  1)  2  36 x 2  24 x  4  2  36 x 2  24 x  6
Practice Problems
If f ( x)  3x 2  2 and g ( x)  2 x  3 find the following:
1.
( f  g )(3)
2.
( f  g )(2)
3.
( g  f )(0)
4.
( f g )(2)
5.
( g ( f (1))
6.
f ( f (2))
7.
g ( g (0))
8.
 f 
  ( x)
g
9.
f ( g ( x))
10. g ( f ( x))
Inverse Relations and Functions
Recall that a relation is a two variable relationship. The inverse of a relation is a relation
that maps the outputs of the original function back to the inputs of the original function. In an
inverse relation its domain is the range values of the original function and its range is the
domain values of the original functions.
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Example: If “A”= (2,1),(3,0),(2,3),(0, 2) then the inverse of
A1 = (1, 2),(0,3),(3, 2),(2,0) . If you were to graph A and A 1 the points of A 1 would be
the reflection of the points of “A” over the line y = x.
Finding the inverse rule for a function:
To find the inverse rule of a function, you can do so by switching the “x” and “y” and
then solve for “y”.
Ex:
y  x 2  3 Find the inverse rule for the given function.
x  y2  3
x  3  y2
y   x3
Ex:
y  3 x  5 Find the inverse rule for the given function
x  3y  5
3y  x  5
x5
y
3
Determining if An inverse function is also a function:
In order to determine if an inverse is also a function you can apply a horizontal line test
to the graph of the original function. If the original function passes the horizontal line test (a
horizontal line only intersects the graph in one place) this is an indication that the inverse of
that function is also a function and the domain and range of the inverse will be the domain and
range of the original function switched. Functions whose inverses are also functions are said
to be one to one functions.
Examples:
Determine if the graphed function has an inverse that is also a function. If the inverse exists
graph it on the same graph.
1.
2.
3.
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Composition of Inverse Functions
If two functions f and f 1 are inverses of each other, then the composite of those
functions: ( f
f 1 )( x)  x and  f
f 1  ( x)  x . What this means is that anytime functions
are inverses of each other the composite of inverse functions always leave you with the
original input.
Example:
f ( x)  3x  4 and f 1 ( x) 
x4
 x4
then f ( f 1 ( x))  3 
4  x44  x
3
 3 
Practice:
Find the inverse of the given relation and say whether or not the inverse is a function.
1.
A  (1, 2),(0,1),(1,0),(2, 1)
2.
y  3x 2  5
3.
y  4x  3
4.
Determine if the graphed function has an inverse that is a function. If it is graph it on the
same coordinate plane.
5.
6..
7.
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