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Chapter 2:
2.1 Functions: definition, notation
A function is a rule (correspondence) that assigns to each element x of one set , say X, one and only one
element y of another set, Y.
The set X is called the domain of the function and the set of all elements of the set Y that are associated
with some element of the set X is called the range of the function. For a function described by above diagram, the domain is the set X= {1,2,3,4} and the range is the set
{a,b,c}.
The elements of a set X are often called inputs and the elements of the set Y are called outputs and
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 inputs x that return an output.
The range of a function is the set of all outputs y.
We will deal with functions for which both domain and the range are the set (or subset) of real numbers
Most often functions in mathematics are defined by a formula or by a graph.
Very often we define a function by giving a formula or an equation which specifies how the y is
obtained when x is given.
Example: The equation y = 3x +2 defines a function since we can easily find the output y when the input,
x, is given. To obtain the y that correspond to a given x, multiply x by 3 and add 2 to the result. For example, if x is 1 then y =3(1) +2 = 5. 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 corresponds to the input 1, that is f(1) = 3∙(1)+2= 5.
If we say that f(4) = -5, then this means that when the input, x, is 4, the output, y , is -5.
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 for -1< x < 1. For example when x = 0, then there are two values of y (± 1) that correspond
to x = 0. Hence, this equations does not describe a function.
Example: Determine whether the equation 2x + 4y^2 -5 = 0 defines y as a function of x. Solution : Let’s solve the equation for y:
2
2
y x
y x
y x
Since there are two solutions, this equation does not define a function
The graph of a function is the set of all points (x, f(x)), where x belongs to the domain of f.
A point (a,b) is on the graph of a function f , if and only if b = f(a).
If f is a function and a is in its domain, then there is one and only one value b that corresponds to a.
Therefore, there is only one point on the graph of f that has a as x-coordinate. This leads us to the
Vertical Line Test
Theorem: If every vertical line crosses the graph at most at one point, then the graph represents 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.
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 f is a function and x is the input then we denote the corresponding
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.
Example: Let f(x) = 3x^2 -2x – 4
Find f(0), f(1), f(-2), f(-x), -f(x), f(x+1), f(2x), f(x+h)
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
Example: For the function x
x g x
( ) , find g(-3), g(0), g(2), g(-x), g(x-2), g(3x), g(x+h).
Solution: 2 5
g
g
g this value is not defined; 2 does not belong to the domain of g
x
x x
x g x
x
x x
x x
x g x
x
x x
x g x 2 3
x
x h x h
x h gx h
Simplifying the difference quotient h
f ( x h ) f ( x )
Example :
Find h
f^ (^ x h ) f ( x )( the difference quotient for f) for f(x) = 3x^2 -2x - 4
6 3 2 3 6 3 2 3 6 3 2 6 3 2
( ) ( ) 3 ( ) 2 ( ) 4 ( 3 2 4 ) 3 ( 2 ) 2 2 4 3 2 4
2 2 2 2
2 2 2 2 2
x h h
h x h h
xh h h h
x xh h h x
h
x xh h x h x x h
x h x h x x h
f x h f x
Example :
Find h
g^ (^ x h ) g ( x )( the difference quotient for g) for x
g x
Solution:
( 2 )( 2 )
3 ( 2 )( 2 )
1 3 ( 2 )( 2 )
1 6 3 6 3 3
( 2 )( 2 )
2 1 3 (^2 )^3 (^2 )
3 2
3 2
3 2 ( )
3 ( ) ( )
x h x x h x
h x h x h
x x h h
x h x
x x h h h
x h x h
x h x h
gx h gx
Domains of functions
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. Often it is more practical to ask what values of x cannot be used in a formula.
At this point, to find a domain of a function you should examine the formula for the function and ask two questions A) Does the formula have a square root (or even index radical)? IF yes, we have to make sure that the expression inside the radical ( called radicand) is not negative. For example, if f ( x ) 3 x 1 then we cannot allow 3x-1 to be negative, as the square root of a negative number does not exist. and
B) Does the formula contain a variable in the denominator? If yes, we have to guarantee that the denominator does not become 0. For example, if
2
x x
x f x , then we can allow any value for x as long x^2 +2x -1 is not zero since the
division by zero is not defined.
If the formula contains a variable in the denominator, the domain consists of all real numbers except the ones that make the denominator zero.
This means that 3x > 1 Dividing both sides by 3 we get : x > 1/ (ii) Write the domain Df [ 1 / 3 ,)
Sometimes the formula for a function might contain both the even index radical and a variable in the denominator or might not contain any of these.
Example: Find the domain of 1 3
2
x
x f x
Solution: Here we have a variable in the denominator and even index radical. Therefore, we should make
sure that the denominator is not zero: (^) x 1 3 0 , and that the radical is well defined: (^) x 1 0.
First we solve x 1 3 0 x 1 3 x 1 32 (after squaring both sides) x 8
Since we want x 1 3 0 , x cannot be 8, x 8 Second , we solve the inequality x 1 0 x 1
Therefore, f(x) will be well defined if x 1 and x 8. This means that Df [ 1 , 8 )( 8 ,).
2.2 The graph of a function- general properties
The graph of a function f is the set of all points (x,y), where x belongs to the domain of f and y = f(x). If a is in the domain of f then the point (a, f(a)) is on the graph of f.
Example: Let f(x) = x^2 – 2. a) If x = 1, what point is on the graph of f? If x = 1 then y = f(1) = 1^2 -2 = -1. Therefore , the point (1,-1) is on the graph of f. b) Is the point (2,-1) on the graph of f? Point (2,-1) would be on the graph of f is -1 = f(2). But f(2) = 2^2 -2 = 2. Thus, the point (2,-1) is not on the graph. c) Sketch the graph of f To sketch the graph of f we need to plot the points (x,f(x)), where x is in the domain of f. We can select some values for x, find corresponding y, plot the points (x,y) and join them with a continuous curve in the order of increasing x values
Table:
x y= f(x) = x^2 -2 (x,y) -3 (-3)^2 -2 = 7 (-3,7) -2 4-2= 2 (-2,2) -1 1-2= -1 (-1,-1) 0 0-2= -2 (0,-2) 1 1-2= -1 (1,-1)
Graph:
A graph of an equation is the graph of a function if it passes the Vertical Line Test : If every vertical line crosses the graph at most at one point, then the graph represents a function.
.
A function not a function
Graphs of some functions:
Constant function : f(x) = C, where C is a constant Example: Graph f(x) = 2
Square root function f(x) = (^) x. It’s domain is the interval [0, +)
The graph of this function is the graph of the equation y = (^) x. We construct the table, plot the points and connect them with a continuous curve.
Reading the graph of a function: domain, range, intercepts
If the graph of a function is given then we can determine its domain, range, x- and y- intercepts as well as find values f(x) for a specific value of x. The domain of a function given by a graph is the set of all x, such that for some y, the point (x, y), is on the graph. Or, in other words, if the vertical line passing through x crosses the graph at some point. The rang e of a function is the set of all y, such that, for some value x, the point (x,y) is on the graph. Or, in other words, if the horizontal line passing through y crosses the graph at some point.
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).
x (^) y = x
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. The y-intercept is a point where the graph crosses or touches the y- axis. 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, f(-2), f(4) and the 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. Similarly, we can find that f(4) = 3. The graph crosses x-axis at (-1,0), so the x-intercept is (-1,0). The y-intercept is (0,1)
Example: Explain why the graph below represents a function. Find its domain and range. Find values of this function when x = -3, -1, 8. List the intercepts.
Solution: The graph represents a function because it passes the Vertical Line Test. Any vertical line crosses the graph at most at one point. Domain = (-7, +∞)
Example: Find the intervals on which the following function is increasing/ decreasing/constant
(i) Identify the parts that are increasing/ decreasing/ constant (ii)
(iii) Determine how the x-coordinate changes when you follow each part, and write it in the interval notation. In the example above, graph is rising (function is increasing) when x- changes from -2 to 2, so f is increasing on (-2,2).
Example: Find the intervals on which the following function is increasing/ decreasing/constant
Solution: We can see that the graph is rising when x changes from -7 to -1 (on the interval (-7,-5)) ; then it is falling when x changes from -5 to -1 (on the interval (-5,-1)); then it rises when x changes from -1 to 1 (on the interval (-1,1)) and finally the graph is falling when x changes from 1 to 6 (on the interval (1,6])
Function is increasing on the intervals (-7, -5), (-1, 1) and decreasing on (-5, -1), (1, 6]
Reading the graph of a function: even, odd functions
A function f is even , if for every x in its domain, -x is also in the domain and f(-x) = f(x). This means that for an even function, if (x,y) is on the graph than (-x,y) is also on the graph. The graph of an even function 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). This means that for an odd function, if (x,y) is on the graph than (-x,-y) is also on the graph. The graph of an odd function is symmetric with respect to the origin. The graph below shows an odd function
A function might be neither odd nor even. The graph below is neither symmetric about the y-axis or the origin.
Using the formula of a function to determine whether a function is odd or even
Knowing whether a function is even or odd helps determine properties of the graph as well as helps in graphing functions.
We use the definitions to check whether function is odd, even or neither. Here is the procedure
(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.
Example: Check whether f ( x ) 3 x^2 2 x 5 is even, odd or neither
Solution: Let’s start with computing and simplifying f(-x).
f ( x ) 3 ( x )^2 2 ( x ) 5 3 x^2 2 x 5
Comparing f(x) and f(-x), we see that the formulas are not the same, therefore, function f is not even.
We check now whether f is an odd function. We compute and simplify -f(x)
f ( x )( 3 x^2 2 x 5 ) 3 x^2 2 x 5
Comparing f(-x) and – f(x) shows that the formulas are not the same, which means function f is not odd.
Hence f is neither even nor odd. The graph of function f is not symmetric with respect to the y axis nor with respect to origin.
Reading the graph of a function: determining intervals on which f(x) > 0, f(x) < 0
Sometimes we can use the graph of a function to solve an inequality of the form f(x) > 0, f(x) > 0, f(x) < 0, f(x) < 0. Note that if f(x) > 0 then value y= f(x) is positive, which means that the point (x, y) is above the x-axis. If f(x) < 0, then the point (x, y) will lie below the x-axis. So, to find all values of x for which f(x) >0 (or (f(x) < 0) ( or to solve the inequality f(x) >0) we must determine which part(s) of the graph is(are) above (below) the x-axis and determine values of x that generate this graph.
Example: The graph of the function f(x) = x^3 + x^2 – 6x is given below. Find values of x for which f(x) >0. (or, solve the inequality x^3 + x^2 – 6x > 0)
(i) identify the parts of the graph that are above the x-axis
(ii) determine values of x that generate the part of the graph with a desired property.
f(x) > 0 when x is in the interval (-3, 0) or (2, +∞). Note that the endpoints are not included, because f(-3) = 0, f(0) = 0, f(2) = 0 and we want f(x) to be strictly greater than 0.
Similarly we can obtain the graphs of the functions below
(B) f(x) = x^3 cube function
(C) f (x) =^3 x cube root function
(D) f(x) =|x| absolute value function
(E) f(x) = x
reciprocal function
Note that the domain of this function is the set of all real numbers except x = 0. Therefore, the graph will consist of two parts- one for x < 0 and one for x > 0. Notice also that if the value of |x| is large, its reciprocal will be close to 0 and if the value of x is close to 0, its reciprocal will have large absolute value, as can be seen in the table below. Therefore, the graph will come closer and closer to the x axis as x becomes larger and larger (positive or negative) and will come closer and closer to the y-axis as x approaches zero. We say that the x – axis is a horizontal asymptote and that the y- axis is a vertical asymptote for the graph of this function.
x y = x
x y = x
