Honors Pre-Calculus Test #2 (Chapters 6-9) Review Problems
1. The graphs below shows two line segments one from (0,0) to (2,6) and the other from (2,6) to (5,0).
For a value of x, with 0 < x < 5, the corresponding y value represents the height from the x-axis to the line
segment, as depicted below.
(a) Find the multipart rule for the height, y, in terms of x for 0 < x < 5.
(b) If a vertical line is drawn (as shown) at x (where 0 < x < 5), give the multipart rule, in terms of x, for
the area of the region to the left of x and bounded by the line segments and the x-axis.
2. Let f(x) = |x + 2| and 𝑔(𝑥) = {
9𝑥 + 3
𝑖𝑓 𝑥 ≤ 1
Write the multipart rule for the function
−3𝑥 + 15 𝑖𝑓 𝑥 ≥ 1
h(x) = 2f(x) - g(x)
3. Ophelia has 300 meters of fencing with which to make two enclosures next to a river. The enclosures will be
rectangular, but she will use the river for one side of each enclosure, so no fencing is needed for that side.
She wants the two enclosures to have the same width, and have one enclosure be twice as long as the other
enclosure. An example of how she might do this is shown in the figure.
What should the dimensions of the smaller enclosure be to make the total combined area of the two
enclosures as large as possible?
4. Let
f(x) = x2 − |x| and g(x) = |x − 2|.
(a) Write the multipart rule for the function h(x) = f(x) + g(x).
(b) Does f(x) have an inverse? If so, find the multipart rule for it. If not, explain why not.
5. Gloria is selling tickets to her glockenspiel concert, and she is trying to decide what the ticket price should
be. She knows from past experience that the number of tickets she will sell is a linear function of the ticket
price. If the ticket price is $2, she will sell 1000 tickets. If the ticket price is $17.34, she will sell 200 tickets.
What should Gloria set the ticket price at for her to make the most money?
Honors Pre-Calculus Test #2 (Chapters 6-9) Review Problems
6. Trapezio Pizzeria sells pizza in the shape of a trapezoid as shown below.
Suppose you make a cut through the pizza, parallel to the right edge and x cm from the
left endpoint. Express the area of the pizza to the left of the cut as a multipart function of x.
7. (a) If f(x) = ax + 5 and f(f(x)) = 4x − 5, then what is the value of a?
(b) Find the inverse of f(x) = x2 − 4x where the domain is restricted to x > 2.
8. Danny has 950 feet of fencing to make a rectangular enclosure with two parallel interior fences that divide
the enclosure into three rectangular regions. What dimensions will maximize the area of the entire
enclosure?
9. A number k is said to be a fixed point for a function f(x) if f(k) = k.
1
(a) Find all fixed points of the function 𝑓(𝑥) = |2 𝑥 − 4|
(b) Let f(x) = 3x − 1 and g(x) = 5x + 8. Find all fixed points of the function h(x) = f(g(x)).
10. Rose is selling tickets to her upcoming tuba performance. She knows from experience that the number of
tickets she sells will be a linear function of the price that she sets for the tickets. If she sets the price at $20,
she will sell 150 tickets. If she sets the price at $18.60, she will sell 162 tickets. At what price will she make
the most money?
11. Joey has a rectangular pizza that is 1 foot by 2 feet long. Joey wants to cut the pizza by making a straight
slice through one corner of the pizza. With a coordinate system imposed as shown in the figures below, we
can represent the slice by the line y = mx. Joey plans to eat the portion of the pizza below the slice (the
shaded region in the figures below). Two possible slices are shown here:
The area of the shaded region depends only on the slope of the slice.
(a) Find the area of the shaded region when the slice has slope m = 0.1.
(b) Find the area of the shaded region when the slice has slope m = 1.3.
(c) Express the area of the shaded region as a multi-part function of the slope m.
12. You decide to take a walk. You start walking due North at 5 feet per second. After walking 200 feet, you
turn and walk due West at 4 feet per second for 30 seconds. You then turn and walk due South at 3 feet per
second for 100 seconds. Express the (straight-line) distance from your starting point as a multipart function
of t, the number of seconds since you started walking.