Business Mathematics I - LI, Lecture 9, December 11, 2013
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1.1
Functions and graphs
Graphical representation
Cartesian coordinate system is formed of two real lines, one horizontal and one vertical, which cross through
their origins. These two lines are called the horizontal axis and the vertical axis.
Every point is represented by two numbers - coordinates. The first number represents the value on axis x and
the second number represents the value on axis y.
We already know that linear function y = ax + b can be graphically represented by a straight line. If we find two
points on the line, we can draw the whole straight line. E.g. the linear function y = 3x + 1 is represented by the
line
1.2
Basic definitions
A function is a relation between a given set of elements (called the domain) and another set of elements (called
the range) such that to each element in the first set there corresponds exactly one element in the second set.
In general we use the term function (denoted by f ) to describe the relationship between independent variable x
and dependent variable y and we write y = f (x). The domain is then the set of numbers that are permitted to
replace the independent variable x. The image of the value of x is the value of y into which the value x is mapped.
The range is a set of all values that the variable y can take.
There are many ways how to represent or visualize functions: a function may be described by a formula or rule,
by a plot or graph, by an algorithm that computes it, or by description of its properties. In applications, functions
are frequently given by tables of values or by formulas.
Example: Determine whether each set defines a function. If it does, then state the domain and range.
1. Cubic function: C = {(−2, −8), (−1, −1), (0, 0), (1, 1), (2, 8)}
Domain = {−2, −1, 0, 1, 2}; Range = {−8, −1, 0, 1, 8}
2. Quadratic function: Q = {(−2, 4), (−1, 1), (0, 0), (1, 1), (2, 4)}
Domain = {−2, −1, 0, 1, 2}; Range = {0, 1, 4}
3. Square root: S = {(0, 0), (1, −1), (1, 1), (4, −2), (4, 2), (9, 3), (9, −3)}
is not a function
Vertical line test: a test which takes a vertical line at any horizontal position. If at any position there is a crossing
at more than one point, then the relation is not a function.
Functions defined by equations: Decide whether y is a function of x: 1. √
3x − 2y = 4; 2. y 2 = x + 1.
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Solving for the dependent variable y: 1. y = −2 + 2 x is a function; 2. y = ± x + 1 does not define a function.
1.3
Inverse function
If f is a function from X to Y then an inverse function for f , denoted as F −1 , is a function from Y to X, with
the property that f −1 (y) = x if f (x) = y. Indeed f −1 = {(y, x)|(x, y) ∈ f }. If f −1 exists, then Domain of f −1 =
Range of f and Range of f −1 = Domain of f .
Example: f converts the temperature C in degrees Celsius to degrees Fahrenheit F , i.e. F = f (C) = 95 C + 32. The
function converting degrees Fahrenheit do degrees Celsius is is equal to f −1 : C = f −1 (F ) = 95 (F − 32).
Horizontal line test: Not every function has an inverse function. The function has an inverse function if each
horizontal line intersects its graph in at most one point. Graphically, inverse function is a mirror image with respect
to the identity function y = x.
Find the inverse function of the function f : 1. Find the domain of f and apply the horizontal line test. 2. solve the
equation y = f (x) for x. The result is an equation in the form x = f −1 (y). 3. Interchange x and y. This expresses
f −1 as a function of x. 4. The domain of f −1 must be the same as the range of f .
Example: Find the inverse of the function y = 3x + 1.
x = y−1
3
y = x−1
3
Examples: Find the inverse of the function if: 1. y =
√
1. y = x + 1 =⇒ x = y 2 − 1 −→ y = x2 − 1
2. y = ln x2 =⇒ x = 2ey −→ y = 2ex
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3. y = e 3 −2 =⇒ x = 3(ln y + 2) −→ y = 3(ln x + 2)
1.4
Combining functions
• Sum function: (f + g)(x) = f (x) + g(x),
√
1
x + 1; 2. y = ln x2 ; 3. y = e 3 −2
• Difference function: (f − g)(x) = f (x) − g(x),
• Product function: (f g)(x) = f (x)g(x),
• Quotient function: ( fg )(x) =
f (x)
g(x) ,
g(x) ̸= 0.
Given the functions f from X to Y and g from Y to Z, then the composition g ◦ f is defined as z = g(y), where
y = f (x), i.e. z = g(f (x)). The domain of g ◦ f is the set of all real numbers x in the domain of f where f (x) is in
the domain of g.
Examples: If f (x) = x2 + 3 and g(x) = 3x − 1 then
(f + g)(x) = f (x) + g(x) = x2 + 3 + 3x − 1 = x2 + 3x + 2
(f + g)(3) = 32 + 3 × 3 + 2 = 20
(f ◦ g)(x) = f (g(x)) = (3x − 1)2 + 3 = 9x2 − 6x + 1 + 3 = 9x2 − 6x + 4
(g ◦ f )(x) = g(f (x)) = 3(x2 + 3) − 1 = 3x2 + 8
(f ◦ g)(2) = 9 × 22 − 6 × 2 + 4 = 28
(g ◦ f )(3) = 3 × 32 + 8 = 35
√
√
Examples: Find (f ◦ g)(x) and its domain for f (x) = 4 − x2 and g(x) = 3 − x.
Domain of f : x2 ≤ 4 ⇐⇒ x ∈ [−2, 2]
Domain of g: x ≤ 3 ⇐⇒ x ∈ (−∞, 3] √
√
√
√
√
(f ◦ g)(x) = f (g(x)) = f ( 3 − x) = 4 − ( 3 − x)2 = 4 − (3 − x) = 1 + x, where x ∈ (−∞, 3] ∩ [−1, ∞) =
[−1, 3].