Chapter 2: More on Functions
Section 2.1: Increasing, Decreasing and Piecewise Functions; Applications.
1. f is increasing on an interval I if for all a, b on the interval a < b implies f (a) < f (b).
f is decreasing on an interval I if for all a, b on the interval a < b implies f (a) > f (b).
2.
List, in interval notation, where the function
is:
(a) Increasing
(b) Decreasing
(c) Constant
3.
List the:
(a) Relative Maximum
(b) Relative Minimum
1
4. Elena and Thomas drive away from a restaurant at right angles to each other. Elena’s speed is 35
mph and Thomas’ is 40mph. Express the distance between the cars as a function of time, d(t) and
find the domain of the function.
5. Jenna’s restaurant supply store has 20 ft of dividers with which to set off a rectangular area for
the storage of overstock. If a corner of the store is used for the storage area, the partition need only
form two sides of a rectangle.
(a) Express the floor area of the storage space as a function of the length of the partition.
(b) Find the domain of the function.
(c) Find the dimensions that maximize the floor area. (Section 2.4)
2
6. Given the function

 x + 1 x < −2
5
−2 ≤ x ≤ 3
f (x) =
 2
x
x>3
evaluate and simplify each of the following
(a) f (−5)
(b) f (−3)
(c) f (0)
(d) f (3)
(e) f (4)
(f) f (10)
(g) Graph!
7. Graph
g(x) =
1
x
3
+3 x<3
−x
x≥3
3
Section 2.2: The Algebra of Functions
1. Sums, Differences, Products and Quotients of Functions
(f ± g)(x) = f (x) ± g(x)
(f g)(x) = (f · g)(x) = f (x)g(x) = f (x) · g(x) (Note: this is different than (f ◦ g)(x)).
f
f (x)
(x) =
, provided g(x) 6= 0.
g
g(x)
Domain:
The domain of f ± g and f g is the intersection of f and g.
The domain of f /g is the intersection of f and g and excludes values of g(x) = 0.
2. The Difference Quotient: (will be on exam/final and will be provided)
f (x + h) − f (x)
h
3. Let f (x) = x + 1 and g(x) =
√
x+3
(a) (f + g)(x), and what is the domain?
(b) (f + g)(6)
(c) (f + g)(−4)
1
4. Let f (x) = x2 + 4 and g(x) = x + 2
(a) (f + g)(x), and domain
(b) (f − g)(x), and domain
(c) (f g)(x), and domain
(d) (f /g)(x), and domain
(e) (gg)(x), and domain
2
5. Find the difference quotient of the following functions:
(a) f (x) = 2x − 3
(b) f (x) = 2x2 − x − 3
(c) f (x) =
1
2x
3
Section 2.3: The Composition of Functions
1. Motivation: You want to convert Fahrenheit to Kelvin. You know:
C(t) = 59 (t − 32) where t is degrees Fahrenheit, C(t) is Celsius
K(t) = t + 273 where t is Celsius and K(t) is Kelvin
What is the temperature in Kelvin if it is 80◦ F?
C(80) = 95 (80 − 32) =
5
9
· 52 = 28.9◦ C
K(28.9) = 28.9 + 273 = 301.9K
What you ended up doing, in a condensed form is: K(C(80)) = (K ◦ C)(80) a composition of
functions!
Instead of plugging in numbers twice, you could set up a function that converts Fahrenheit to
Kelvin directly.
5
5
5
2297
(K ◦ C)(t) = K(C(t)) = K
(t − 32) =
(t − 32) + 273 = t +
9
9
9
9
(Do you see what steps we skipped? Fill in details if you need to!)
2. The composition of functions:
(f ◦ g)(x) = f (g(x)),
where x is in the domain of g and g(x) is in the domain of f (x).
3. Given f (x) = 2x − 5 and g(x) = x2 − 3x + 8, find the following:
(a) (f ◦ g)(7)
(b) (f ◦ g)(x)
1
f (x) = 2x − 5 and g(x) = x2 − 3x + 8
(c) (g ◦ f )(7)
(d) (g ◦ f )(x)
(e) (g ◦ g)(x)
(f) (f ◦ f )(x)
2
4. Let f (x) =
√
x and g(x) = x − 3
(a) What is (f ◦ g)(x) and domain
(b) What is (g ◦ f )(x) and domain
5. Let f (x) =
1
5
and g(x) =
x−2
x
(a) What is (f ◦ g)(x) and domain
(b) What is (g ◦ f )(x) and domain
3
6. Let h(x) = (2x − 3)5 find f (x) and g(x) such that h(x) = (f ◦ g)(x)
7. Let h(x) =
1
find f (x) and g(x) such that h(x) = (f ◦ g)(x)
(2x − 3)2
4
Section 2.4: Symmetry:
1. Example of points that are symmetric with respect to (wrt) the x-, y- axis, and origin
2. Algebraic Tests of Symmetry:
x−axis: If replacing y with −y produces an equivalent equation, then the graph is symmetric
with respect to the x−axis.
y−axis: If replacing x with −x produces an equivalent equation, then the graph is symmetric
with respect to the y−axis.
origin: If replacing x with −x and y with −y produces an equivalent equation, then the graph
is symmetric with respect to the origin.
3. Test y = x2 + 2 for symmetry:
4. Test y 4 + x2 = 15 for symmetry:
1
5. Graphic example of even/odd functions:
6. Even Functions and Odd Functions:
Even: If the graph of a function f is symmetric with respect to the y−axis, we say it is an even
function. That is, for each x in the domain of f , f (x) = f (−x)
Odd: If the graph of a function f is symmetric with respect to the origin, we say it is an odd
function. That is, for each x in the domain of f , f (−x) = −f (x)
7. Are the functions even, odd or neither?
(a) f (x) = 5x7 − 6x3 − 2x
(b) g(x) = 5x6 − 3x2 − 4
(c) h(x) = x2 − x
2
Section 2.6: Variation and Applications
1. Definitions: Let k > 0 be a constant (variation constant or constant of proportionality)
1. Direct Variation: y = kx, y varies directly as x or y is directly proportional to x
2. Inverse Variation: y = xk , y varies inversely as x or y is inversely proportional to x
3. Combination:
(a) y = kxn , y varies directly as the nth power of x or y is directly proportional to xn
(b) y =
k
,
xn
y varies inversely as the nth power of x
(c) y = kxz, y varies jointly as x and z.
2. Find the variation constant and an equation of variation in which y varies directly as x and y = 32
when x = 2.
3. The number of centimeters of water W produced from melting snow varies directly as S, the
number of centimeters of snow. Meteorologists have found that under certain conditions 150 cm of
snow will melt to 16.8 cm of water. To how many centimeters of water will 200 cm of snow melt
under the same conditions?
4. Find the variation constant and an equation of variation in which y varies inversely as x and
y = 16 when x = 0.3.
1
5. The time t required to fill a swimming pool varies inversely as the rate of flow r of water into the
pool. A tank truck can fill a pool in 90 minutes at a rate of 1500L/min. How long would it take to
fill the pool at a rate of 1800L/min?
6. The volume of wood V in a tree varies jointly as the height h and the square of the girth g. (Girth
is distance around.) If the volume of a redwood tree is 216m3 when the height is 30m and the girth
is 1.5m, what is the height of a tree whose volume is 960m3 and whose girth is 2m?
7. The volume V of a given mass of a gas varies directly as the temperature T and inversely as the
pressure P . If V = 231 cm3 when T = 42◦ and P = 20kg/cm2 , what is the volume when T = 30◦ and
P = 15kg/cm2 ?
2