Pre-calculus 12
Combinations and Compositions of functions – and Even and Odd functions
In class, we looked at combining two (or more) functions in a variety of ways: adding, subtracting, multiplying, dividing, and
composing. Composing two functions is like inserting one inside of the other. We have two specific notations for indicating a
composition:
f  g  x   or
f
g  ( x)
which reads “F of G of x” and indicates that function g(x)
was inserted into function f(x). That is, the output from g is the input to f
You should give a careful examination when considering the domain of combined or composed functions.
Special properties of function composition
f ( x)  f ( x) , then f(x) is considered an even function. However, if –x
is inserted into f(x) and the result is such that f ( x)   f ( x) , then f(x) is considered an odd function. The properties of
If –x is inserted into f(x) and the result is such that
even and odd functions are summarized in the chart.
Note: A function can be even, odd, or neither.
Algebraic property by inserting -x
Geometric property on graph
Example of a graph
y = x2
Graph is symmetrical about the y-axis
f (  x)  f ( x)
Even
y = x3
o
Graph is rotated 180 about the origin
(or graph is reflected in the x then yaxis)
f (  x)   f ( x)
Odd
1. Given that
a)
f ( x)  x  5 , g ( x )  x  3 , h ( x ) 
f ( x)  g ( x)
b)
h( x)  g ( x)
x
, find the domain of each:
x 1
c)
g  f ( x) 
d)
2. You might want two different colored pencils. Complete the sketch of the graph (at
right) such that it represents
a) an even function
b) an odd function
3. For each, insert –x into the function and determine if the function is even, odd, or
neither.
a)
f ( x)  x 3  2 x
b)
g ( x)  x 6  3 x 2  1
c)
h( x)  3 x
d)
j ( x)  x 2  4 x
e)
k ( x)  2 x
f)
m( x ) 
1
x
h  f ( x) 
e)
g  h( x ) 
4. Use the graph to the right to evaluate the following:
** each grid line is 1 unit
a)
 f  g  3
b)
 f  g  0
c)
f  g  4  
d)
g  f  4
e)
f  f  1 
f)
f g  g  2

f(x)

g(x)
Intro to inverse functions
Inverse functions can be thought of as another special application of composition: if two functions f(x) and g(x) are composed
such that f ( g ( x))  g ( f ( x))  x , then we can say that f(x) and g(x) are inverses of each other. The notation we use to
indicate that two functions are inverses:
f ( x) and f 1 ( x) . Note that f 1 ( x) 
1
f ( x)
. A function may not have a true
inverse, or it may have an inverse only if its domain is restricted. Since the output of one function can be thought of the input
into the inverse (and vice versa), we can think of inverse functions has having “their x and y values swapped.” Therefore, the
geometric property of the graphs of inverse functions is that they are reflections of each other across the line y = x. Always
remember: inverse functions undo each other.
1. a) Graphically, what must be true about a function f(x) for it to have an inverse f -1(x)? (“what must be true for an inverse
of a function to be a function itself?”
b) therefore, which of these – if any – have an inverse?
i)
iv) y = x4 – 6x + 12
c)
ii)
v)
y x
vi)
iii)
y xx
for the graphs in i-iii, sketch each graph as it is reflected across the line y = x, then restrict the range (if
required) of this new graph so that it is a function.
2. Sketch the inverse of the function at right 
3. What is the inverse of y = x3? Is this function always the inverse?
4. If
g ( x)  x 2  4 x and x  2 , find g 1 (5)
5. If
f ( x)  3x  7 and g ( x)  3 x , find  f 1 g 1  (2) [this is the same as f 1  g 1  2   ]
6. Determine which of these are inverses of each other
p( x) 
1
x 1
q ( x)  1 
1
x 1
r ( x)  1 
1
1 x
s ( x) 
1
1
x
7. Algebraically determine the inverse of each function (TRY restricting the domain/range as necessary (so that our function
was 1-to-1), so it might be necessary to see the graphs of each/their inverses):
1
x2
a)
f ( x) 
d)
i ( x)  x  2
g ( x) 
e)
j ( x)  x 3  5
8. a) Sketch a fairly accurate depiction of
b) State the domain and range of
c) Find
1  3x
5  2x
b)
f  x 
f  x
f 1  x  , sketch this graph
d) State the domain and range of
f 1  x 
e) State any conclusions you can draw from this.
x 1
x2
c)
h( x)  ( x  6)2 , x  3
f)
k ( x)  x 2  4 x  7