Mathematics for Business and Economics - I Chapter 2 – Functions and Graphs

2.1 Functions The idea of a function is this: a correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R. The first set is called the domain, and the set of corresponding elements in the second set is called the range. Notation: if y is a function of x, we write y = f(x) Other common symbols for functions include but are not limited to g, h, F, G

Function Evaluation Consider our function What does f (-3) mean? Replace x with the value –3 and evaluate the expression The result is 7 . This means that the point (-3,7) is on the graph of the function.

Let g be the function defined by the equation y = x2 – 6x + 8. EXAMPLE 1 Evaluating a Function Let g be the function defined by the equation y = x2 – 6x + 8. Evaluate each function value. Solution

EXAMPLE 1 Evaluating a Function Solution continued

AGREEMENT ON DOMAIN If the domain of a function that is defined by an equation is not explicitly specified, then we take the domain of the function to be the largest set of real numbers that result in real numbers as outputs.

Domain of a Function Consider which is not a real number. Question: for what values of x is the function defined? Answer: is defined only when the radicand (3x-2) is equal to or greater than zero. This implies that Therefore, the domain of our function is the set of real numbers that are greater than or equal to 2/3.

Example1: Find the domain of the function Answer: Example : Find the domain of In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except 5/3.

Find the domain of each function. EXAMPLE 3 Finding the Domain of a Function Find the domain of each function. Solution a. f is not defined when the denominator is 0. Domain: {x|x ≠ –1 and x ≠ 1}

EXAMPLE 3 Finding the Domain of a Function Solution continued The square root of a negative number is not a real number and is excluded from the domain. Domain: {x|x ≥ 0}, [0, ∞) The square root of a negative number is not a real number and is excluded from the domain, so x – 1 ≥ 0. However, the denominator ≠ 0.

Domain: {x|x > 1}, or (1, ∞) EXAMPLE 3 Finding the Domain of a Function Solution continued So x – 1 > 0 so x > 1. Domain: {x|x > 1}, or (1, ∞) Any real number substituted for t yields a unique real number. Domain: {t|t is a real number}, or (–∞, ∞)

Find the domain of f(x) = x/(x2 –x – 2) Example #2a (p.80) Find the domain of f(x) = x/(x2 –x – 2) The domain would be the set of all real numbers except those values of x which set the denominator equal to zero These values are found by factoring (x2 –x – 2) = (x + 1)(x - 2) x = -1, 2 So the domain is the set of all real numbers , except x =-1, 2

Equality of Functions Two functions, f and g are equal (f = g) if The domain of f is equal to the domain of g For every x in the domain of f and g, the values of the two functions are the same; that is f(x) = g(x)

Example #1 (p. 79-80) Which of the following functions are equal f(x) = (x + 2)(x + 1)/(x – 1) g(x) = x + 2 h(x) = x + 2 Domains of g, h, the set of all real numbers and are equal, but the domain of f is the set of all real numbers except x = 1

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2 A polynomial function of degree n is a function of the form where n is a nonnegative integer and the coefficients an, an–1, …, a2, a1, a0 are real numbers with a ≠ 0.

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2.3 Combinations of Functions (i) Sum (ii) Difference (iii) Product (iv) Quotient (v)

EXAMPLE 1 Combining Functions

f(x) = x2 , g(x) = 3x, find; f(x) + g(x) = x2 + 3x f(x).g(x) = 3x3 EXAMPLE 2 Combining Functions f(x) = x2 , g(x) = 3x, find; f(x) + g(x) = x2 + 3x f(x).g(x) = 3x3 f(x) – g(x) = x2 – 3x f(x)/g(x) = x2/3x = x/3 cf(x) = cx2

Find each of the following functions. EXAMPLE 3 Combining Functions Let Find each of the following functions. Solution

EXAMPLE 3 Combining Functions Solution continued

EXAMPLE 3 Combining Functions Solution continued

COMPOSITION OF FUNCTIONS If f and g are two functions, the composition of function f with function g is written as and is defined by the equation where the domain of values x in the domain of g for which g(x) is in the domain of f. consists of those

COMPOSITION OF FUNCTIONS

Find each of the following. EXAMPLE 1 Evaluating a Composite Function Let Find each of the following. Solution

EXAMPLE 1 Evaluating a Composite Function Solution continued

Find each composite function. EXAMPLE 2 Finding Composite Functions Let Find each composite function. Solution

EXAMPLE 2 Finding Composite Functions Solution continued

EXAMPLE 3 Finding the Domain of a Composite Function Solution

Domain is (–∞, 0) U (0, ∞). Domain is (–∞, –1) U (–1, ∞). EXAMPLE 3 Finding the Domain of a Composite Function Solution continued Domain is (–∞, 0) U (0, ∞). Domain is (–∞, –1) U (–1, ∞).

following provides a decomposition of H(x). EXAMPLE 4 Decomposing a Function Show that each of the following provides a decomposition of H(x).

EXAMPLE 4 Decomposing a Function Solution

An ordered pair of real numbers is a pair of real numbers in which the order is specified, and is written by enclosing a pair of numbers in parentheses and separating them with a comma. The ordered pair (a, b) has first component a and second component b. Two ordered pairs (x, y) and (a, b) are equal if and only if x = a and y = b. The sets of ordered pairs of real numbers are identified with points on a plane called the coordinate plane or the Cartesian plane.

Definitions We begin with two coordinate lines, one horizontal (x-axis) and one vertical (y-axis), that intersect at their zero points. The point of intersection of the x-axis and y-axis is called the origin. The x-axis and y-axis are called coordinate axes, and the plane formed by them is sometimes called the xy-plane. The axes divide the plane into four regions called quadrants, which are numbered as shown in the next slide. The points on the axes themselves do not belong to any of the quadrants.

Definitions The figure shows how each ordered pair (a, b) of real numbers is associated with a unique point in the plane P, and each point in the plane is associated with a unique ordered pair of real numbers. The first component, a, is called the x-coordinate of P and the second component, b, is called the y-coordinate of P, since we have called our horizontal axis the x-axis and our vertical axis the y-axis.

Definitions The x-coordinate indicates the point’s distance to the right of, left of, or on the y-axis. Similarly, the y-coordinate of a point indicates its distance above, below, or on the x-axis. The signs of the x- and y-coordinates are shown in the figure for each quadrant. We refer to the point corresponding to the ordered pair (a, b) as the graph of the ordered pair (a, b) in the coordinate system. The notation P(a, b) designates the point P in the coordinate plane whose x-coordinate is a and whose y-coordinate is b.

Graph the following points in the xy-plane: EXAMPLE 1 Graphing Points Graph the following points in the xy-plane: Solution 3 units right, 1 unit up 2 units left, 4 units up 3 units left, 4 units down 2 units right, 3 units down 3 units left, 0 units up or down

Solution continued EXAMPLE 1 Graphing Points Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

PROCEDURE FOR FINDING THE INTERCEPTS OF A GRAPH Definitions The points where a graph intersects (crosses or touches) the coordinate axes are of special interest in many problems. Since all points on the x-axis have a y-coordinate of 0, any point where a graph intersects the x-axis has the form (a, 0). The number a is called an x-intercept of the graph. Similarly, any point where a graph intersects the y-axis has the form (0, b), and the number b is called a y-intercept of the graph. PROCEDURE FOR FINDING THE INTERCEPTS OF A GRAPH Step1 To find the x-intercepts of an equation, set y = 0 in the equation and solve for x. Step 2 To find the y-intercepts of an equation, set x = 0 in the equation and solve for y.

EXAMPLE 1 Finding Intercepts

Step 1 To find the x-intercepts, set y = 0, solve for x. EXAMPLE 2 Finding Intercepts Find the x- and y-intercepts of the graph of the equation y = x2 – x – 2. Solution Step 1 To find the x-intercepts, set y = 0, solve for x. The x-intercepts are –1 and 2.

Step 2 To find the y-intercepts, set x = 0, solve for y. EXAMPLE 2 Finding Intercepts Solution continued Step 2 To find the y-intercepts, set x = 0, solve for y. The y-intercept is –2.

The graph of the linear equation(line) The following steps can be used to draw the graph of a linear equation. Step1 ) Select at least 2 values for x Step2 ) Substitute them in the equation and find the corresponding values for y Step3 ) Plot the points on cartesian plane Step4 ) Draw a straight line through the points.

X -2 -1 1 y 3 5 5 3 1 -2 -1 1 -1

x-intercept y=0  0=2x+3 x= 2/3 (2/3 , 0) is x-intercept y- intercept x=0  y=2.0+3 y= 3 (0,3) y-intercept y-intercept x-intercept

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Step2 Step3 Step4 x-intercept x-intercept 3/2 -2 5 y-intercept Vertex point -49/4

Step1 Step3 Step2

Step4 Step5

Ex : Sketch the Graph of

2 4 Vertex point y-intercept -8