College Algebra 9th txtbk

SECTION 3–5 Operations on Functions; Composition 223
3-5 Operations on Functions; Composition
Z Performing Operations on Functions
Z Composition
Z Mathematical Modeling
Perhaps the most basic thing you’ve done in math classes is operations on numbers: things
like addition, subtraction, multiplication, and division. In this section, we will explore the
concept of operations on functions. In many cases, combining functions will enable us to
model more complex and useful situations.
If two functions f and g are both defined at some real number x, then f(x) and g(x) are
both real numbers, so it makes sense to perform the four basic arithmetic operations with
f(x) and g(x). Furthermore, if g(x) is a number in the domain of f, then it is also possible
to evaluate f at g(x). We will see that operations on the outputs of the functions can be used
to define operations on the functions themselves.
Z Performing Operations on Functions
The functions f and g given by
f(x) 2x 3 and g(x) x 2 4
are both defined for all real numbers. Note that f(3) 9 and g(3) 5, so it would seem
reasonable to assign the value 9 5, or 14, to a new function ( f g)(x). Based on this
idea, for any real x we can perform the operation
f(x) g(x) (2x 3) (x 2 4) x 2 2x 1
Similarly, we can define other operations on functions:
f(x) g(x) (2x 3) (x 2 4) x 2 2x 7
f(x)g(x) (2x 3)(x 2 4) 2x 3 3x 2 8x 12
For x 2 (to avoid zero in the denominator) we can also form the quotient
f(x)
g(x) 2x 3
x 2 4
x 2
Notice that the result of each operation is a new function. So, we have
( f g)(x) f(x) g(x) x 2 2x 1
( f g)(x) f(x) g(x) x 2 2x 7
( fg)(x) f(x)g(x) 2x 3 3x 2 8x 12
a f f(x)
b(x)
g g(x) 2x 3
x 2 x 2
4
Sum
Difference
Product
Quotient
The sum, difference, and product functions are defined for all values of x, as were the original
functions f and g, but the domain of the quotient function must be restricted to exclude
those values where g(x) 0.