This document provides examples of simplifying expressions involving functions. It defines several functions, including f(x)=2-3x, g(x)=-2x, and h(x)=(2x-1)/(x-2). It then gives examples of combining these functions using operations like addition, subtraction, multiplication, division, and composition. It also gives word problems translating situations into functions that represent things like the cost of boxes of peanuts or cashews as a function of the number of boxes, or the area of an expanding circle city as a function of years.

1. Notation and Algebra of Functions
Exercise. A. Simplify the following expressions with the given
functions.
2 + 3x – 1
2x – 1
f(x) = 2 – 3x g(x) = –2x h(x) = x–2
1. f(2) + f(3) 2. 2f(3) 3. 2g(0) + g(1) 4. [h(2)]2
5. f(0) + g(0) + h(0) 6. 3h(1) – g(–2 )
7. 3h(1) – g(–2 ) 8. f(2)/3 + f(3)/2
9. 2f(–3) – 2g(–1) 10. [f(3)]2 – [g(3)]2
11. f(0) + g(0) + h(0) 12. [f(3) – g(3)]2
13. h(1) / h(–1 ) 14. [h(1/2)]2
15. [f(1/2)]2 16. [g(1/2)]2
17. g(f(0)) 18. f(g(0)) 19. g(h(0))
20. h(g(0)) 21. f(h(0)) 22. h(f(0))

2. Notation and Algebra of Functions
Exercise. B. Simplify the following expressions
with the given functions.
x–1
f(x) = 3 + 2x g(x) = –x2 + 3x – 2 h(x) = x–2
23. f(2a) 24. g(2a) 25. 2g(a) 26. h(2a)
27. 2h(a) 28. f(3 + b) 29. g(3 + b) 30. h(3 + b)
31. f(3 + b) – f(b) 32. g(3 + b) – g(b) 33. h(3 + b) – h(b)
34. f(3 + b) – f(3 – b) 35. g(3 + b) – g(3 – b)
36. g(x) + 3f(x) 37. 2g(x) + [f(x)]2 38. g(x) / h(x)
39. a. Peanuts cost $9.00/box, what is the cost of x boxes of peanuts?
b. Cashews cost $12.00/box, what is the cost of x boxes of cashews?
c. Let x = the number of boxes in one order. We have coupons for $7
off for one order of x boxes of peanuts.
What is the cost P(x) for x boxes peanuts with the coupon?

3. Notation and Algebra of Functions
c. Let x = the number of boxes in one order. There is a surcharge
(special tax) of $5 per cashew–order for x boxes of cashews. What is
the cost C(x) for an order of x boxes cashews?
d. Let x = the number of boxes in one order.
Simply 2P(x) + 3C(x). What does this function represent?
40. Recall that the area of a circle is A = π * r2.
A circle city of radius r = 5 km is expanding outwardly with the
radius of the city increasing at a rate of 2 km every year.
Let x = the number of years,
a. what is the radius r(x) of the city after x years?
b. after 10 years, what is r(10)?
what is the area when x = 10?
c. what is the area A(x) of the city expanding
(2 km/yr) r=5 km
after x years?
d. what is A(6)? A(8)? A(8) – A(6)?
What does each expression mean?