1.3 Creating New Polynomial Functions: An Introduction to ...

Johann Peter Gustav
Lejeune Dirichlet
(1805–1859)
Originally, the
definition of a function
was much less precise
than it is today.
Dirichlet was
responsible for the
definition of a function
that we use today.
21. Application: A solar panel is used to power an overhead sign on a highway.
Each square metre of solar panel receives about 200 W of solar power. This
solar panel converts about 15% of the solar energy to electrical power.
(a) Write a function for the solar power, S(A), measured in watts (W) for
any given area, A, in square metres.
(b) Write a function for the total electrical power, P(S), in watts (W) this
solar panel can generate.
(c) How large must the solar panel be if the sign uses 6 W of electrical
energy
22. Refrigeration slows down the growth of bacteria in food. The number of
bacteria in a certain food is approximated by B(T) 15T 2 70T 600,
where T represents the temperature in degrees Celsius and 3 ≤ T ≤ 12.
Once the food is removed from refrigeration, the temperature, T(t),
is given by T(t) 3.5t 3, where t is the time in hours and 0 ≤ t ≤ 3.
(a) Write the expression for the number of bacteria in the food, t hours
after it is removed from refrigeration.
(b) At 1.5 h, about how many bacteria are in the food
(c) When will the bacteria count reach about 1200
23. A franchise owner operates two coffee shops. The sales, S 1
, in thousands of
dollars, for shop one are represented by S 1
(t) 700 1.4t 2 , where t 0
corresponds to the year 2000. Similarly, the sales for shop two are
represented by S 2
(t) t 3 3t 2 500.
(a) Which shop is showing an increase in sales after the year 2000
(b) Determine a function that represents the total sales for the two
coffee shops.
(c) What are the expected total sales for the year 2006
(d) If sales continue according to the individual functions, what would you
recommend that the owner do Explain.
C
24. Thinking, Inquiry, Problem Solving: Let f (x) mx 2 2x 5 and
g(x) 2x 2 nx 2. The functions are combined to form the new function
h(x) f (x) g(x). Points (1, 40) and (1, 24) satisfy the new function.
Determine f (x) and g(x).
25. Let h(k) 3k 2, g(t) 3t 2, and f (x) 3x 2.
(a) Determine C(k) f (g(h(k))).
(b) Show in a diagram how k 4 carries through to its final value.
1.3 CREATING NEW POLYNOMIAL FUNCTIONS: AN INTRODUCTION TO COMPOSITION 37