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3.1 Functions
A relation is a set of ordered pairs (x, y).
Example: The set {(1,a), (1, b), (2,b), (3,c), (3, a), (4,a)} is a relation
A function is a relation (so, it is the set of ordered pairs) that does not contain two pairs with the same
first component. Sometimes we say that a function is a rule (correspondence) that assigns to each
element of one set , X, one and only one element of another set, Y. The elements of the set X are often
called inputs and the elements of the set Y are called outputs. A function can be visualized as a machine, that takes x as an input and returns y as an output.
The domain of a function is the set of all first components, x, in the ordered pairs.
The range of a function is the set of all second components, y, in the ordered pairs.
We will deal with functions for which both domain and the range are the set (or subset) of real numbers
A function can be defined by:
(i) Set of ordered pairs Example: {(1,a), (2,b), (3,c), (4,a)} is a function, since there are no two pairs with the same first component. The domain is then the set {1,2,3,4} and the range is the set {a,b,c} Example: {(1,a), (2,b), (1,c), (4,a)} is not a function, since there are two pairs with the first component 1 (ii) Diagram which shows how the elements of two sets are paired.
1 is paired with a , so (1,a) belongs to function The domain is the set X= {1,2,3,4} and the range is the subset {a,b,c} of Y.
(iii) A formula or equation which specifies how the y is obtained when x is given.
Example: y = 3x +2 defines a function since we can easily find the output y when the input, x, is given. If x is 1 then y =3(1) +2 = 5, for example. So every x has a corresponding y value.
We say that a function is given explicitly by an equation, if the equation is of the form
y = (formula that contains only x variable). We say that a function is given implicitly , if it is given by an equation that is not in the above form. For example, y = 3x + 2 is a function given in an explicit form, and 3x - y + 2 = 0 is the same function given in an implicit form.
When we use a formula to define a function we often give a name to a function(f,g,h,etc.) and use a special notation for the output, y. If x is the input, then we denote the output by f(x) (read “f of x”). Caution: f(x) is not a multiplication of f and x. It is an entity that can’t be split. f(x) denotes output that corresponds to the input x. For example, instead of writing y = 3x+2, we often write f(x) = 3x+2. With this notation f(1) denotes the output (y) that correspond to the input 1, that is f(1) = 3∙(1)+2= 5. If we say that f(4) = -5, that means that when the input is 4, the output is -5. This also means that the function f contains a pair (4, -5)
When a function f is given by a formula and its domain is not given, then it is assumed that the domain is the largest set of real numbers for which f (x) can be computed and is a real number. An equation defines a function if it can be solved for y and the solution is unique.
Example: 3x+2y = 4 is a function, because we can solve it for y, y = (-3/2)x + 2 and the solution is unique. The equation x^2 + y^2 = 1 is not a function, because when we solve it for y, we get y 1 x^2 , two solutions.
(iv) The graph, which is the set of all pairs (x,y) that are plotted in the coordinate system. The graph is the set of all points (x, f(x)), where x belongs to the domain of a function f. A point (a,b) is on the graph of a function f , if and only if b = f(a) A graph represents a function, if it passes the Vertical Line Test , which says that If any vertical line crosses the graph at most at one point, then the graph is that of a function. The domain of a function given by a graph is the set of all x -coordinates of the points on the graph. The range is the set of all y-coordinates of the points on the graph
Example: Find f(0), f(1), f(-2), f(-x), -f(x), f(x+1), f(2x), f(x+h) for f(x) = 3x^2 -2x – 4
f(0) = 3(0)^2 -2(0) – 4 = - f(1) = 3(1)^2 – 2(1) – 4 = - f(-2) = 3(-2)^2 -2(-2)- 4= 12 f(-x) = 3(-x)^2 -2(-x) -4 = 3x^2 + 2x – 4 f(x+1) = 3(x+1)^2 -2(x+1) – 4 = 3(x^2 + 2x +1)-2x-2 -4 = 3x^2 + 6x + 3 -2x – 6= 3x^2 + 4x – 3 f(2x) = 3(2x)^2 -2(2x) – 4 = 3(4x^2 ) – 4x – 4= 12x^2 – 4x – 4 f(x+h) = 3(x+h)^2 -2(x+h) – 4 = 3(x^2 + 2xh +h^2 ) – 2x- 2h -4 = 3x^2 + 6xh + 3h^2 -2x – 2h – 4
- f(x) = - (3x^2 – 2x – 4) = -3x^2 + 2x + 4
(ii) positive 0 negative 0 positive
-2 4 (iii) Df ( , 2 ][ 4 ,)
C) Does the formula contain a logarithm? We will deal with such functions in Chapter 6
Remarks: 1) If you answer no to questions A) and B) then the domain of the function is the set of all real numbers
- If you answer yes to two of these questions, you must follow the procedure for each question and the domain of the function will be the intersection of all sets obtained in each question.
The intersection of two sets is the common part of those two sets, that is the set that contains numbers that belong to both sets. Example: The intersection of the red and green intervals, is the blue interval
3. 2 The graph of a function
The graph of a function f is the set of all points (x,f(x)), where x belongs to the domain of f Example:
Since the point (a, b) is on the graph then b = f(a )
The graph of an equation is the graph of a function if it passes the Vertical Line Test. That is if every vertical line crosses the graph at most at one point.
A function not a function
The domain of a function given by a graph is the set of all x, such that (x, y), for some y, is on the graph. The rang e of a function is the set of all y, such that the point (x,y), for some x, is on the graph.
To find value f(a) from the graph, locate the point on the graph whose x-coordinate is a. The y- coordinate of that point is f(a). The x-intercepts are the points where the graph crosses or touches the x- axis. They are found by solving the equation f(x) = 0. They are the points at which f has value zero, and therefore they are often called the zeros of function f. To find y-intercept, compute f(0). A function can have at most one y-intercept.
Example: The graph of a function f is given below. Find the domain and the range. Find f(-2). Find x- and y-intercepts , if any.
If you follow the graph from left to right, the x – coordinate changes from - to 6, so Df = (-, 6]. At the same time, the y-coordinate changes from - to 4, so Range of f= (-, 4]. To find f(- 2), find the point on the graph with x-coordinate -2. This point is (-2,-7). The y-coordinate of this point is -7, hence f(-2) = -7. The graph crosses x-axis at (-1,0), so the x-intercept is (-1,0). The y-intercept is (0,1)
A function f is even , if for every x in its domain, -x is also in the domain and f(-x) = f(x). If f is even, then its graph is symmetric with respect to the y-axis. The graph below shows an even function.
A function f is odd , if for every x in its domain, -x is also in its domain and f(-x) = -f(x). If f is an odd function, its graph is symmetric with respect to the origin. The graph below shows an odd function
To check whether a function is odd/even/neither follow the following steps:
(i) Evaluate f(-x) and simplify (ii) Compare f(-x) with f(x). If the formulas are the same then f is even. If the formulas are not the same, then f is not even. (iii) Write the formula for – f(x) (iv) Compare f(-x) with – f(x). If the formulas are the same then f is odd. If the formulas are not the same then f is not odd.
Remarks: i) If f is even, then it cannot be odd and vice versa
ii) There are functions that are neither odd nor even
Example: Check whether 1
x
x f x is even, odd or neither
(i) 1
x
x x
x f x
(ii) f(x) and f(-x) are not the same, so f is not even
(iii) 1
x
x x
x f x
(iv) f(-x) and – f(x) are the same, so f is odd. Hence, the graph of f(x) is symmetric with respect to the origin.
3.4 Library of functions; Piecewise defined functions
Here are the graphs that you must be able to recognize and draw without making the table.
f(x) = x identity function f(x) = x^2 square function
f(x) = x^3 cube function f(x) = x square root function
To find f(a), you must first determine whether a < 1 or a > 1. If a < 1, then you will use the first
formula, f(x) = 2x +3. If a > 1, then you will use the second formula, f(x) = |x|. For example, to
evaluate f(-3), we first notice that -3 < 1, so we use the formula reserved for x < 1, which is
f(x) = 2x +3.
Hence, f(-3) = 2(-3)+ 3 = -3.
To graph a piecewise function : graph each function involved in the formula and take the piece of that
graph that corresponds to the given values of x.
Example: The graph of f(x) from previous example is in red in the picture below. The dashed graphs of
y = 2x+3 and y = |x| should be erased.
3. 5 Graphing techniques
Graphs of certain functions can be obtained by graphing one of the functions in the library and performing some transformations on it. Suppose c is a positive real number, c > 0. If the graph of a function f is given then to obtain the graph of (i) g(x) = f(x) + c (g(x) = f(x) – c) shift the graph of f up (down) by c units. The graph of g will have the same shape as that of f but it will be in a different place. Each point on the graph of f will be moved vertically up (down). If (x,y) is on the graph of f(x) then (x, y + c) ((x,y-c) ) will be on the graph of g.
(ii) g(x) = c f(x) stretch the graph of f vertically, if c > 1 and compress vertically when 0 < c <1. If the point (x,y) is on the graph of f then (x, c y) is on the graph of g. The x-intercepts are not affected by this transformation.
-8 -6 -4 -2 2 4 6 8
2
4
6
8
x
y
(2,4)
(2,1)
down 3 units
y = x^2
(-1,-2) y = x (^2) -
(-1,1)
-8 -6 -4 -2 2 4 6 8
2
4
6
8
x
y
(1,1)
(4,4) (-3,3)
(1,1/2)
(-3,3/2)^ (4,2)
y = f(x)
y = 1/2 f(x)
-8 -6 -4 -2 2 4 6 8
2
4
6
8
x
y
(1,1)
(4,2)
(1,3)
(4,6)
y = f(x)
y = 3 f(x)
(v) g(x) = f(x + c) (g(x) = f(x-c) ) Shift the graph of f horizontally to the left ( right) by c units. Each point on the graph of f will be moved horizontally c units to the left ( right). If (x,y) is on the graph of f then (x-c, y) ((x +c, y)) will be on the graph of g.
(vi) g(x) = f(cx) compress the graph of f horizontally, if c > 1 and stretch it horizontally if 0 < c < 1. If (x,y) is on the graph of f then (1/c x, y) will be on the graph of g
The transformations can be combined and performed in a sequence. If present, shift up/ down MUST be performed last.
Example: Use transformations to graph f(x) = -3 x 1 2 You must determine first the order of transformations
(5) f(x) = -3 x 1 + 2 A constant (2) is added to y = -3 x 1. This means that we need to graph y = -3 x 1 and move it 2 units up
(4) y = - 3 x 1 Putting a minus sign in front of a function results in reflection about x-axis We need to graph to y = 3 x 1 and reflect it about x-axis
(3) y = 3 x 1 x 1 is multiplied by 3, so we have a vertical stretch. It is enough to graph y = x 1 and stretch it by a factor of 3
(2) y = x 1 replacing x- 1 with x, gives us y = x , which is a basic function. To get the graph of y = x 1 we must shift the graph of y = x one unit to the right
(1) y = x
Once you end up with a basic function, stop and start graphing in the reverse order: 5), 4), 3), 2) 1)
