Eng. Math 1 by W. Mukendi
2020
Chapter 1: Functions and Models
1.1 Four ways to represent a function
A function f is a rule that assigns to each element x in a set D exactly one element,
called 𝑓(𝑥𝑥), in a set E.
The set D is called the domain of the function. The number f(x) is the value of f at x
and is read “f of x”. The range of f is the set of all possible values of f(x) as x varies
throughout the domain.
Independent variable is a symbol that represents an arbitrary number in the domain
of a function f.
Dependent variable is a symbol that represents a number in the range of f.
Functions arise whenever one quantity depends on another. Consider the following
three situations:
a. The area A of a circle depends on the radius r of the circle. The rule that
connects r and A is given by the equation = 𝜋𝑟 2 . With each positive number
r there is one associated value of A, and we say that A is a function of r.
b. The human population of the world P depends on the time t. the table gives
estimates of the world population P(t) at time t, for certain years. For instance
, 𝑃(1950) ≈ 2,560,000,000 . for each value of the time t there a corresponding
value of P, and we say that P is a function of t.
Year 1900 1910 1920 1930 1940 1950 1960 1970 1980
Population 1650 1750 1860 2070 2300 2560 3040 3710 4450
(millions)
c. The cost C of mailing an envelope depends on its weight w. the post office
has a rule for determining C when w is known.
Each of these examples describes a rule whereby, given a number (r, t or w)
another number (A, P or C) is assigned. In each case we say that the second
number is a function of the first number.
1.1.1 Representations of Functions
There are four possible ways to represent a function:
• Verbally (by a description in words)
• Numerically (by a table of values)
• Visually (by a graph)
• Algebraically (by an explicit formula).
1.1.2 Examples
Page 1 of 13
, Eng. Math 1 by W. Mukendi
2020
1) Sketch the graph and find the domain and range of each function:
a) 𝑓(𝑥𝑥) = 2𝑥𝑥 − 1 b) 𝑔(𝑥𝑥) = 𝑥𝑥 2
𝑓(𝑎+ℎ)−𝑓(𝑎)
2) if 𝑓(𝑥𝑥) = 2𝑥𝑥 2 − 5𝑥𝑥 + 1 and ℎ ≠ 0, evaluate
ℎ
3) When you turn on a hot-water faucet, the temperature T of the water depends
on how long the water has been running. Draw a rough graph of T as a
function of the time t that has elapsed since the faucet was turned on.
4) A rectangular storage container with an open top has a volume of 10 𝑚3 . The
length of its base is twice its width. Material for the base costs $10 per square
meter; material for the sides costs $6 per square meter. Express the cost of
materials as a function of the width of the base.
5) Find the domain of each function:
1
a) 𝑓(𝑥𝑥) = √𝑥𝑥 + 2 b) 𝑔(𝑥𝑥 ) =
𝑥 2 −𝑥
Note: The following test will let you know which curves in the xy-plane are graphs of
functions.
The vertical line Test “A curve in the xy-plane is the graph of a function of x if
and only if no vertical line intersects the curve more than once”.
Y Y
X x
(a) This curve represents a function (b) this curve doesn’t represent a function.
The truth of the vertical line test can be seen in the figures above. If each vertical
line 𝑥𝑥 = 𝑎 intersects a curve only once, at (a,b), then exactly one function value is
defined by 𝑓(𝑎) = 𝑏. But if a line 𝑥𝑥 = 𝑎 intersects the curve twice, at (a,b) and
Page 2 of 13
2020
Chapter 1: Functions and Models
1.1 Four ways to represent a function
A function f is a rule that assigns to each element x in a set D exactly one element,
called 𝑓(𝑥𝑥), in a set E.
The set D is called the domain of the function. The number f(x) is the value of f at x
and is read “f of x”. The range of f is the set of all possible values of f(x) as x varies
throughout the domain.
Independent variable is a symbol that represents an arbitrary number in the domain
of a function f.
Dependent variable is a symbol that represents a number in the range of f.
Functions arise whenever one quantity depends on another. Consider the following
three situations:
a. The area A of a circle depends on the radius r of the circle. The rule that
connects r and A is given by the equation = 𝜋𝑟 2 . With each positive number
r there is one associated value of A, and we say that A is a function of r.
b. The human population of the world P depends on the time t. the table gives
estimates of the world population P(t) at time t, for certain years. For instance
, 𝑃(1950) ≈ 2,560,000,000 . for each value of the time t there a corresponding
value of P, and we say that P is a function of t.
Year 1900 1910 1920 1930 1940 1950 1960 1970 1980
Population 1650 1750 1860 2070 2300 2560 3040 3710 4450
(millions)
c. The cost C of mailing an envelope depends on its weight w. the post office
has a rule for determining C when w is known.
Each of these examples describes a rule whereby, given a number (r, t or w)
another number (A, P or C) is assigned. In each case we say that the second
number is a function of the first number.
1.1.1 Representations of Functions
There are four possible ways to represent a function:
• Verbally (by a description in words)
• Numerically (by a table of values)
• Visually (by a graph)
• Algebraically (by an explicit formula).
1.1.2 Examples
Page 1 of 13
, Eng. Math 1 by W. Mukendi
2020
1) Sketch the graph and find the domain and range of each function:
a) 𝑓(𝑥𝑥) = 2𝑥𝑥 − 1 b) 𝑔(𝑥𝑥) = 𝑥𝑥 2
𝑓(𝑎+ℎ)−𝑓(𝑎)
2) if 𝑓(𝑥𝑥) = 2𝑥𝑥 2 − 5𝑥𝑥 + 1 and ℎ ≠ 0, evaluate
ℎ
3) When you turn on a hot-water faucet, the temperature T of the water depends
on how long the water has been running. Draw a rough graph of T as a
function of the time t that has elapsed since the faucet was turned on.
4) A rectangular storage container with an open top has a volume of 10 𝑚3 . The
length of its base is twice its width. Material for the base costs $10 per square
meter; material for the sides costs $6 per square meter. Express the cost of
materials as a function of the width of the base.
5) Find the domain of each function:
1
a) 𝑓(𝑥𝑥) = √𝑥𝑥 + 2 b) 𝑔(𝑥𝑥 ) =
𝑥 2 −𝑥
Note: The following test will let you know which curves in the xy-plane are graphs of
functions.
The vertical line Test “A curve in the xy-plane is the graph of a function of x if
and only if no vertical line intersects the curve more than once”.
Y Y
X x
(a) This curve represents a function (b) this curve doesn’t represent a function.
The truth of the vertical line test can be seen in the figures above. If each vertical
line 𝑥𝑥 = 𝑎 intersects a curve only once, at (a,b), then exactly one function value is
defined by 𝑓(𝑎) = 𝑏. But if a line 𝑥𝑥 = 𝑎 intersects the curve twice, at (a,b) and
Page 2 of 13
