Functions and Their Graphs

b. Graphs of Functions and Equations

The graph of a function f is the set of allpoints (x, y) in the xy plane for which (x, y) is an orderedpair in f.From this definition, the graph of f is the same asthe graph of the equation y = f(x).

Example : The graph of the function in Illustration 1 is sketched in Figure 1. This can be done by considering the domain and range of the function without using point plotting.a. Definition of FunctionsA function defined on a set D, called the Domain, is a set of ordered pairs of numbers (x, y) in which no two distinct ordered pairs have the same first number. The number f(x) (read as f of x) is called the function value of the function f at the number x. The set of all admissible values of x is called the Domain of f, and the set of all resulting values y = f(x) is called the Range, denoted by R. The number x is called the independent variable and y is called the dependent variable.X1

X2

X3y1

y2

y3

y4a

b

ce

dNot a functiona functionFigure 1Figure 2In Figure 2, the Domain is the set {a, b, c} and the Range is the set {d, e,} Illustration 1:

Domain of f: Note that for f(x) to be defined, 4 - x2 0. Thus, -2 x 2.Range of f: It is specified that f(x) is the positive square root of 4 - x2 where -2 x 2. This tells us that the range is [0, 2].

Figure 1:Graph of

One can determine whether a set is function or not by considering its graph and use the vertical line test. A vertical line should intersect the graph of a function in at most one point.In Figure 1, a vertical line intersects the graph in at most one point.Vertical Line TestIllustration 2:The set {(x, y) x2 - y2 = 9} is not a function since there corresponds two values of y for one value of x.The graph of this equation is a hyperbola with center at (0, 0) oriented vertically. See the figure below.

The vertical line test shows that this is not a function.

Example 1: Given the function f(x) = x2 + 1. Find the Domain and range of f.

Solution:Domain of f:f is always defined for any real number x. So,D = {x x is any real number or - < x < +} Range of f: We see that x2 0. So, R = {y y 1}a. f(2x) = (2x)2 + 1 = 4x2 + 1b. f(x + h) = (x + h)2 + 1 = x2 + 2xh + h2 + 1

Example 2:Determine the domain and range of the function and sketch the graph.

We see that Example 1 is a quadratic function, 2 is a radical function, 3 is linear function with a hole, 4 is a piecewise-defined function, and 5 is an absolute-value function.These are some examples of types of functions.c. Operations on FunctionsGiven two functions f and g, their sum,difference, product and quotient are definedas follows:(f + g)(x) = f(x) + g(x)(f - g)(x) = f(x) - g(x)(f g)(x) = f(x) g(x)(f / g)(x) = f(x) / g(x)

Example : The functions f and g are defined by f(x) = 1/x2 and g(x) = 4 + x2define the following functions and find the domain of the resulting function: a. f + g , b. f g, c. f gd. f/g

Solution:

The domain of each of the resulting functions (a d) is the set of real numbers except 0.d. Composition of FunctionsThe composition of two functions f and g is the function f g defined by(f g)(x) = f(g(x))where the domain of f g is the set of all numbers x in the domain of g such that g(x) is in the domain of f. This is also called composite function.

e. Types of Functions1. Algebraic Functions An algebraic function is formed by a finite number of algebraic operations on a variable where the function is defined. These algebraic operations include addition, subtraction, multiplication, division, raising powers, and extracting roots.A polynomial function f of degree n is a function of the form f(x) = anxn + an-1xn-1 + an-2xn-2 + + a1x + a0where a0, a1, , an are real numbers, an 0 and n is a nonnegative integer. The function f(x) = 2x4 6x3 + 8x 7 is a polynomial function of degree 4. 1. A constant function is a function whose range consists of only one number. It is denoted by f(x) = c where c is a constant. Its graph is a horizontal line at a directed distance of c units from the x-axis. The graph of the constant function f(x) = 3 is shown in the figure .Kinds of Polynomial Functions:

y = 32. A linear function is a function of the form f(x) = mx + b where m and b are constants and m 0. It is the slope-intercept form of a line where m is the slope and b is the y-intercept. Example: The function f(x) = 2x 3 is linear with m = 2 and b = -3 and whose graph is shown in the figure .

y = 2x -3 The function f(x) = x is particular linear function called the identity function. Its graph is the line bisecting the first and third quadrants.

3. A quadratic function is defined by f(x) = ax2 + bx + c where a, b, and c are constants and a 0. Its graph is a parabola opening upward if a > 0 or opening downward if a < 0 where the vertex is at x =

Example: The function defined by f(x) = 2x2 4x + 5 is quadratic. Its graph is a parabola opening upward with vertex at (1, 3).

Observe that a linear function is a polynomial of degree 1 and a quadratic function is of degree 2. If a polynomial is of degree 3, it is called a cubic function. A rational function is a function that can be expressed as the quotient of two polynomials. Polynomial and rational functions are particular kinds of algebraic functions.Example: f(x) =

Even and Odd Function A function f is an even function if for every x in the domain of f, f(-x) = f(x).ii) A function f is an odd function if for every x in the domain of f, f(-x) = - f(x).Other Types of FunctionsExample(a) The function f(x) = x2 is an even function since f(-x) = (-x)2 = x2. Its graph is a parabola symmetric with the y-axis. See the figure.(b) The function g(x) = x3 is an odd function because g(-x) = (-x)3 = -x3. Its graph is symmetric with respect to the origin. See the figure.

y = x2

y = x3Radical Functions of the form

Piece-defined Functions or Conditional Functions

Absolute-Value Function

242. LIMIT OF A FUNCTION

Definition of Limit of a Function

Theorems on Limit of a Function

One-Sided Limits

Infinite Limits

Limits at Infinity

Continuity of a Function

Trigonometric Limitsxy 25To understand the concept of a limit, let us consider the function:

Let us observe what happens to f(x) when x is made to approach 2 from the right (that is values of x greater than 2) and when x is made to approach 2 from the left (that is values of x less than 2).

Table 1 shows the values of f when x approaches 2 from the right. As x approaches 2 from the right, what do you observe with the values of f?

Table 2 shows the values of f when x approaches 2 from the left. As x approaches 2 from the left, what do you observe with the values of f?

Table 1 Table 2

x151.56.51.97.71.997.971.9997.997

x3112.59.52.18.32.018.032.0018.003

2.1 Limit of a Function, its Definition

Definition of Limit of a Function

Let f be a function defined at every number in some open interval containing a, except possibly at the number a itself. The limit of f(x) as x approaches a is L, written as

if the following statement is true:Given any > 0, however small, there exists a > 0 such that if 0< x - a < then f(x) - L < .

26y = f(x)xyaa+1a - 1LL+1L - 1This means that f(x) approaches L as x approaches a if the absolute value of the difference between f(x) and L can be made as small we please by taking x sufficiently near a but not equal to a.27Example:Show by definition that

Solution:Let > 0. We want to find such thatif 0< x - 4 < then 2x +1 - 9 = 2x - 8 < .If 0< x - 4 < then 2x +1 - 9 < .if 0< x - 4 < then 2x - 8 < .if 0< x - 4 < then 2 x - 4 < .if 0< x - 4 < then x - 4 < /2. So, we take = /2. , 282.2 Theorems on Limits of FunctionsTheorem 1 (Limit of Identity Function)If f(x) = x, then Example:

Theorem 2 (Limit of a Constant Function)If c is a constant, then for any number a,

Example:Theorem 3 (Limit of a Linear Function)If m and b are any constants,

Example:Theorem 4 (Limit of the Sum of two functions)If and , then Example:

29Theorem 5 (Limit of the Product of two functions)If and , then

Example: Theorem 6 (Limit of the Quotient of two functions)If and such that M 0, then

Example:

Theorem 7 (Limit of the nth Power of a function) If and n is any positive integer, then

Example:Theorem 8 (Limit of the nth root of a function)If n is a positive integer and then

Example: Note that if n is even, L should be greater than 0.

Evaluate the following limits30Answer:364. 7. -1/310. 2/5 13. 1/6-155. 78. 011. -3214. -1/92/36. 09. 612. -1/415. 1/4

For nos. 7 15, you need to simplify the function (by factoring or by conjugate) before you can find the limit. You do this when by initial substitution, the result is 0/0.Solution to selected numbers:

8. 11.

10.12.

322.3 One Sided-LimitsDefinition 2.1Let f be a function that is defined at every number in some open interval (a,c), then the limit of f(x) as x approaches a from the right is L written as .

This is the right-hand limit.

Definition of Right-Hand Side Limit

Definition 2.2

Let f be a function that is defined at every number in some open interval (a,c), then the limit of f(x) as x approaches a from the left is M written as .

This is the left-hand limit.

Definition of Left-Hand Side Limit

Theorem: The two sided limit exists and is equal to L if and only if and both exist and are both equal to L.

33

Example: Use the graph of y = f(x) to find the limit of the following:1. 3. 5. 7. 9.2. 4. 6. 8.

34Example: (p.57 58)Find the indicated limit:

#18.

#32. Given .

Find the values of a and b such that and

both exist.

Answer: a) 0c) does not existe) 1 b) 1d) 1f) 1Answer: and

Solving for a, b in a b 3 = 0 and 3a + b + 15= 0We get a = -3 and b = -6352.4 Infinite Limits

What do you observe in the function values reflected in the two tables?36Let f be a function that is defined at every number in some interval I containing a, except possibly at the number a itself. As x approaches a, f(x) increases without bound and is written as

Definition of Infinite Limit, Increasing without Bound

Let f be a function that is defined at every number in some interval I containing a, except possibly at the number a itself. As x approaches a, f(x) decreases without bound and is written as

Definition of Infinite Limit, Decreasing without Bound

37

38

Example: Find the limits of the following functions.

40Definition of a Vertical Asymptote

2.5 Limits at Infinity

What do you observe in the function values reflected in the two tables?41Let f be a function defined at every number in some interval (a,+), the limit of f(x) as x increases without bound is L, written as

Definition of Limit at Infinity, x Increasing without Bound

Let f be a function defined at every number in some interval (a,+), the limit of f(x) as x decreases without bound is L, written as

Definition of Limit at Infinity, x Decreasing without Bound

Limits at Infinity

?

Continuity

D e r i v a t i v e s

Differentiation Formulas

Product and Quotient Rules