KEY
Name: _________________________________
Precalculus (A)
Homework 1.4
Questions 1 & 2: Find the indicated information for each function, f(x), below.
(-∞, ∞)
Domain: _______________________________
1)
y ∈ ℝ (all real numbers)
Range: ________________________________
(-∞, -2) x < -2
Solutions to the inequality f(x) < 0: ________________________
(2, 6)
Any maxima: _________________________
(ordered pair(s))
(6, 2)
Any minima: __________________________
(ordered pair(s))
(-∞, 2) ∪ (6, ∞)
Increasing on: ________________________________
(intervals)
(2, 6)
Decreasing on: _______________________________
(intervals)
(-2, 0), (0, 3)
(-2, 0)
Inflection point(s): ________________ (ordered pair(s))
Concave up on: _____________________________(intervals)
(-∞, -2) ∪ (0, 6) (intervals)
Concave down on: ______________________
∞
lim f (x) = ________
x →∞
lim f (x) = _________
-∞
x →−∞
Average rate of change on the interval [-2, 4]: _________________
(-∞, ∞)
Domain: _______________________________
2)
y ≤7
Range: ________________________________
(-7, 7) & (3, 5)
Any maxima: _________________________
(-1, 1)
Any minima: __________________________
(-∞, -7) ∪ (-1, 3)
Increasing on: ________________________________
(-7, -1) ∪ (3, ∞)
Decreasing on: _______________________________
(-4, 4) & (1, 3)
Inflection points: ________________
(-4, 1)
Concave up on: ______________________________
(-∞, -4) ∪ (1, ∞)
Concave down on: ______________________
-∞
lim f (x) = ________
Average rate of change on the interval [-4, 1]: ________
x →∞
-∞
lim f (x) = _________
x →−∞
Questions 3 – 5: Find the average rate of change for f(x) on the given interval.
3) f(x) = 3x2 – 8x + 2; [4, 8]
4) f ( x) =
x −3
; [5, 12]
x
5) f (x) = x + 8 ; [-4, 1]
28
Questions 6 & 7: Draw the graph of a function with each set of characteristics.
6) infinite discontinuity at x = -2
increasing on (-∞, -2)
increasing on (-2, ∞)
f(-6) = -6
7) continuous
average rate of change on [3, 6] is 4
decreasing on (6, ∞)
f(3) = -4
(6, 8)
(3, -4)
(-6, -6)
8) Which is greater, the average rate of change of f (x) = x2 on the interval [0, 1] or on the interval [4, 5]?
Avg. rate of
Avg. rate of
change = 1
change = 9
9) If the average rate of change of f(x) on the interval [a,b] is positive, is f(x) sometimes, always, or never increasing on
[a,b]? Explain your reasoning.
f(x) is sometimes increasing on the interval [a, b]. For the average rate of
change to be positive on [a, b], it matters only that f(b) > f(a).
10) What is the slope of the secant line from (a,f(a)) to (b, f(b)) when f(x) is constant for the interval [a,b]? Explain your
reasoning. The slope of the secant line is equal to 0. If the function is constant on that
interval, then f(a) = f(b), so subtracting those values gives you 0.