Transcript The Algebra of Functions - St. Edward's University

The Algebra of Functions
• Functions can be combined by the usual operations of
addition, subtraction, multiplication, and division. Note that
the domain of the combined function may not be the same as
that of either of the original functions.
• Example. Let f(x) = x2 and g(x) = x  1. The functions f + g,
f  g, f ·g, and f /g are defined as follows:
( f  g )(x)  f ( x)  g ( x)  x 2  x  1
( f  g )(x)  f ( x)  g ( x)  x 2  x  1
( f  g )(x)  f ( x)  g ( x)  x 3  x 2
f
f ( x)
( x) 

g
g ( x)
x2
.
x 1
Composite Functions
• If we take the output of function f and use it as the input to
function g, we will have defined a new function h which is
called the composite of g and f , denoted g  f .
• The new function h is defined by
h( x)  ( g  f )(x)  g[ f ( x)]
which is read “g of f of x”. The domain of g  f may be
smaller than the domain of f (do you see why?).
• Example. Let f(x) = x2 and g(x) = x1.
f [ g ( x)]  f ( x  1)  ( x  1) 2  x 2  2 x  1
g[ f ( x)]  g ( x 2 )  x 2  1.
Note that the two composite functions are not equal.
One-to-one Functions
• An element in the range of a function may have more than one
preimage. If we require that every element of the range has
only one preimage, then the function is called one-to-one.
• More formally,
A function f is one- to - oneif f (a)  f (b)  a  b.
• To determine whether a function is one-to-one, we can use the
horizontal line test, which is:
If no horizontal line meets the graph of a function in more
than one point, then the function is one-to-one.
Problems for One-to-one Functions
• Which of the following function graphs is the graph of a oneto-one function?
Inverse Functions
• Suppose that f is a one-to-one function. Then f has an inverse
function, f -1 and
f 1 ( y)  x if and onlyif y  f ( x).
• Example. Find the inverse of f(x) = 2x  3. Simply solve for x
in terms of y.
2x  3  y
2x  y  3
y3
x
2
y3
1
f ( y) 
.
2
Verification of Inverse Functions
• For f and f -1, the two composite functions must satisfy
f 1 ( f ( x))  x, for all x in domain of f
f ( f 1 ( y))  y, for all y in range of f .
• Example. Verify the above relations for f(x) = 2x  3 and
y3
1
f ( y) 
.
2
(2 x  3)  3 2 x
1
f ( f ( x)) 

x
2
2
 y  3
1
f ( f ( y ))  2
  3  ( y  3)  3  y.
 2 
The Graphs of f and f -1
• Suppose we plot both f and f -1 on the same coordinate axes.
The graphs of f and f -1 are reflections of each other about the
line y = x.
x3
• Example. Plot f(x) = 2x  3 and f 1 ( x) 
on the same
2
coordinate axes.
y = 0.5x+1.5
y=x
y = 2x3
Summary of Algebra of Functions; We discussed
• Addition, subtraction, multiplication, and division of functions
• Composition of functions, which is not commutative
• One-to-one functions and the horizontal line test
• Inverse functions and finding the inverse algebraically
• Verification of inverse functions
• Graphs of f, f -1 and how they relate