Applied Mathematics for Business and Economics

Lecture Note Function

b. Since negative numbers do not have real square roots, the only values of x for

which gx()= x−^2 can be evaluated are those for whichx−^2 is nonnegative,

that is, for whichxxD−≥ ≥20 or 2 or = +∞[2, ).

1.3 Composition of Functions .........................................................................

The composite function gh⎡⎣(x)⎤⎦is the function formed from the two functionsg()u

andhx()by substituting hx()for u in the formula forg(u).

Example 4

Find the composite function ghx⎣⎦⎡⎤( ) if g(uu u)=^2 ++ 31 andhx x( )= + 1.

Solution
Replace u by x+ 1 in the formula for g to get.

(^) () ( ) ( )
(^22)
ghx⎡⎤⎣⎦=++ ++=++x13 11x x 5 x 5
Example 5
An environmental study of a certain community suggests that the average daily level
of carbon monoxide in the air will be Cp( )=0.5p+ 1 parts per million when the
population is p thousand. It is estimated that t years from now the population of the
community will bePt()=+10 0.1t^2 thousand.
a. Express the level of carbon monoxide in the air as a function of time.
b. When will the carbon monoxide level reach 6.8 parts per million?
Solution
a. Since the level of carbon monoxide is related to the variable p by the equation.
Cp( )=0.5p+ 1
and the variable p is related to the variable t by the equation.
Pt()=+10 0.1t^2
It follows that the composite function
CPt⎡⎤⎣⎦()=+ = ++=+C()10 0.1t^22 0.5 10 0.1( t) 1 6 0.05t^2

⎤⎦

expresses the level of carbon monoxide in the air as a function of the variable t.

b. Set CPt⎣⎡()equal to 6.8 and solve for t to get

2

2

2

6 0.05 6.

0.05 0.

16

4

t

t

t

t

+=

=

=

=

That is, 4 years from now the level of carbon monoxide will be 6.8 parts per million.