Functions and graphs

Question 1

What is meant by an ordered pair?

Answer 1

If X and Y are sets, in that generality, and x is an element of X and y is an an element of Y, then we write (x, y) for what we call the ordered pair consisting of x and y in the order displayed.

We accept that the notation for an ordered pair of real numbers is the same as that for an open interval, but the practice is strongly entrenched and almost never causes any confusion. Note that (y, x) is in general different from (x, y)

Question 2

How is the set for all ordered pairs (x, y) denoted?

Answer 2

We will write X x Y for the set of all ordered pairs (x, y), where x is an element of X and y is an element of Y. For example, if X is the set of oil prices and Y is the set of interest rates, then an element of X x Y is a pair (p, r), where p is an oil price and r is an interest rate.

Question 3

What is meant by a relation R in terms of X x Y?

Answer 3

A relation R from a set X to a set Y is a subset of XxY, meaning that any element of R is also an element of X 􏰂 Y.

Question 4

If it happens that (x, y) is an element of R, how do we write it?

Answer 4

x is R-related to y and write xRy

Question 5

If we write p ◦ l, what do we mean?

Answer 5

let P and L denote, respectively, the set of all points and the set of all lines in a given plane. For an ordered pair .p; l/ in P 􏰂 L, it is either the case that “p is on l” or “p is not on l”. If we write p ı l for “p is on l”, then ı is a relation from P to L in the sense of this paragraph.

Question 6

What is meant by a function f from a set X to a set Y?

Answer 6

Function f from a set X to a set Y is a relation from X to Y with the special property that if both xfy and xfz are true, then y = z.

(In many books, it is also required that for each x in X there exists a y in Y, such that xfy. We will not impose this further condition.)

Question 7

What is wrong with the following statement?:

“(Prime) interest rates are a function of oil prices”

Answer 7

With this definition we see that the notion of function is not symmetric in x and y. The notation f:W X –> Y is often used for “f is a function from X to Y” because it underscores the directedness of the concept.

The relation R defined by pRr if “there has been a time at which both the price of oil has been p and the (prime) interest rate has been r” does not define a function from oil prices to interest rates. Many people will be able to recall a time when oil was $30 a barrel and the interest rate was 8% and another time when oil was $30 a barrel and the interest rate was 1%. In other words, both (30, 8) and (30, 1) are ordered pairs belonging to the R relation, and since 8 /= 1, R is not a function.

Lest you think that we may be trying to do it the wrong way around, let us write R° for the relation from the set of interest rates to the set of oil prices given by rR°p if and only if pRr. If you can remember a time when the interest rate was 6% with oil at $30 a barrel and another time when the interest rate was 6% with oil at $70 a barrel, then you will have both .6; 30/ and .6; 70/ in the relation R. The fact that 30 ¤ 70 shows that R is also not a function.

Question 8

If $100 is invested at, say, 6% simple interest, then the interest earned I is a function of the length of time t that the money is invested. These quantities are related by ?

Answer 8

I = 100(0.06)t

Question 9

Why might this be written as

I(t) = 100(0.06)t?

Answer 9

we will often write I(t) = 100(0.06) to reinforce the idea that the I-value is determined by the t-value.

Question 10

Why might we write

I = I(t)?

Answer 10

Sometimes we write I = I(t) to make the claim that I is a function of t even if we do not know a formula for it

Question 11

Give a definition for a function

Answer 11

A function f : X –> Y is a rule that assigns to each of certain elements x of X at most one element of Y. If an element is assigned to x in X, it is denoted by f(x).

Question 12

What is meant by the domain of a function?

Answer 12

The subset of X consisting of all the x for which f(x) is defined is called the domain of f.

Question 13

What is meant by the range of a function?

Answer 13

The set of all elements in Y of the form f(x), for some x in X, is called the range of f.

Question 14

What is meant by an indepedent and dependent variable fo a function?

Answer 14

A variable that takes on values in the domain of a function is sometimes called an input, or an independent variable for f. A variable that takes on values in the range of f is sometimes called an output, or a dependent variable of f. Thus, for the interest formula I = 100(0.06)t, the independent variable is t, the dependent variable is I, and I is a function of t.

Question 15

In the function
y^2 = x + 2
Is y a function of x? explain

Answer 15

If x is 9, then y^2 = 9, so y = ̇3. Hence, to the input 9, there are assigned not one but two output numbers: 3 and -3. This violates the definition of a function, so y is not a function of x.

Question 16

In the function
y = x + 2

let f represent this rule where x = 1

Answer 16

f(1) = 3

f(x) which is read “f of x,” and which means the output, in the range of f, that results when the rule f is applied to the input x, from the domain of f.

Question 17

What are meant by function values?

Answer 17

Outputs are also called function values.

Question 18

Unless otherwise stated, the domain of a function f : X -> Y is the set of all x in X for which f (x) makes sense, as an element of Y. When X and Y are both (-∞, ∞) this convention often refers to what?

Answer 18

Arithmetical restrictions:
in h(x) 1 / x - 6
Here any real number can be used for x except 6, because the denominator is 0 when x is 6. So the domain of h is understood to be all real numbers except 6.

Question 19

Give a notation for any real number can be used for x except 6

Answer 19

(-∞, ∞) - {6}

Question 20

what does “A - B for the set of all x in X” mean?

Answer 20

x is in A and x is not in B.

Question 21

To say that two functions f,g : X -> Y are equal, denoted f = g, is to say what?

Answer 21

1. The domain of f is equal to the domain of g;

2. For every x in the domain of f and g, f(x) = g(x).

Question 22

f(x) = x^2 
g(x) = x^2 for x > 0

f(x) = g(x) ?

Answer 22

f /= g. For here the domain of f is the whole real line (-∞, ∞) and the domain of g is (0, ∞).

Question 23

f(x) = (x + 1)^2 
g(x) = x^2 + 2x + 1

f(x) = g(x) ?

Answer 23

for both f and g, the domain is understood to be (-∞, ∞) and the issue for deciding if f = g is whether, for each real number x, we have (x + 1)^2 = x^2 + 2x + 1. But this is true, it is a special case of item 4 in the Special Products of Section0.4

Question 24

How would you find the domain of the function

f(x) = x / x^2 - x - 2

Answer 24

We cannot divide by 0 so we must find any values of x that make the denominator 0. These cannot be inputs. Thus, we set the denominator equal to 0 and solve for x:

x^2 - x - 2 = 0
(x - 2) (x + 1) = 0
x = 2, -1

Therefore, the domain of f is all real numbers except 2 and -1.

Question 25

What is the domain of g(t) = sqrt(2t - 1)

Answer 25

sqrt(2t - 1) is a real number if 2t - 1 is greater than or equal to 0. If 2t - 1 is negative then sqrt(2t - 1) is not a real number, so we must assume that 2t - 1 >= 0 2t >= 1 t >= 1/2 Thus the domain is the interval (1/2, ∞)

Question 26

g(x) = 3x^2 - x + 5 find g(z)

Answer 26

g(z) = 3z^2 - z + 5

Question 27

g(x) = 3x^2 - x + 5 find g(r^2)

Answer 27

g(r^2) = 3(r^2)^2 - (r^2) + 5 = 3r^4 - r^2 + 5

Question 28

How would you solve if d(x) = x^2 find f(x+ h) - f(x) / h

Answer 28

``` The expression f(x+ h) - f(x) / h is referred to as a difference quotient. Here h, the numerator, is a difference of function values. We have: f(x+ h) - f(x) / h = (x+ h)^2 - x^2 / h = x^2 + 2hx + h^2 - x^2 = 2hx + h^2 / h = h(2x + h) / h = 2x + h for h/= 0 ```

Question 29

Why do we say h/=0?

Answer 29

If we consider the original difference quotient as a function of h, then it is different from 2x + h because 0 is not in the domain of the original difference quotient but it is in the default domain of 2x + h. For this reason, we had to restrict the final equality.

Question 30

What is meant by a demand function?

Answer 30

Suppose that the equation p = 100/q describes the relationship between the price per unit p of a certain product and the number of units q of the product that consumers will buy (that is, demand) per week at the stated price. This equation is called a demand equation for the product. If q is an input, then to each value of q there is assigned at most one output p. that is, when q is 20, p is 5. Thus, price p is a function of quantity demanded, q. This function is called a demand function. Since q cannot be 0 (division by 0 is not defined) and cannot be negative (q represents quantity), the domain is all q > 0.

Question 31

What is meant by a constant function?

Answer 31

Let h : (-∞, ∞) -> (-∞, ∞) be given by h(x) = 2. The domain of h is (-∞, ∞), the set of all real numbers. All function values are 2. For example, h(2) = 2; h(23) = 2; h(781265) = 2 We call h a constant function because all the function values are the same. More generally, a function of the form h(x) = c, where c is a constant, is called a constant function.

Question 32

A constant function belongs to a broader class of functions, called ______, define

Answer 32

A constant function belongs to a broader class of functions, called polynomial functions. In general, a function of the form f(x) = cnX^n + c(n-1)X^(n-1) ..... c1X + c0 where n is a nonnegative integer and are constants with cn /= 0, is called a polynomial function (in x).

Question 33

in f(x) = 3X^2 - 8X + 9, what is the degree and the leading coefficient?

Answer 33

The number n is called the degree of the polynomial, and cn is the leading coefficient. Thus f(x) = 3X^2 - 8X + 9 is a polynomial function of degree 2 with leading coefficient 3.

Question 34

Polynomial functions of degree 1 or 2 are called ____ or ____ functions, respectively.

Answer 34

Polynomial functions of degree 1 or 2 are called linear or quadratic functions, respectively.

Question 35

What is meant by a rational function?

Answer 35

A function that is a quotient of polynomial functions is called a rational function. f(x) = x^2 - 6x / x + 5 is a rational function since the numerator and the denominator are both polynomials.

Question 36

Is g(x) = 2x + 3 a rational function?

Answer 36

Yes since 2x + 3 = 2x + 3 / 1. In fact, every polynomial function is a rational function

Question 37

What is meant by a case-defined function?

Answer 37

F(s) = {1 if - 1 =< s < 1, 0 if 1 =< s < 2, s - 3 if 2 =< s =< 8} This is called a case-defined function because the rule for specifying it is given by rules for each of several disjoint cases. Here s is the independent variable, and the domain of F is all s such that - 1 =< s =< 8

Question 38

How do you combine two fundtions to get a new function?

Answer 38

``` There are several ways of combining two functions to create a new function. Suppose f and g are the functions given by f(x) = x^2 and g(x) = 3x adding these gives f(x) + g(x) = x^2 + 3x or (f+g)(x) = f(x) + g(x) = x^2 + 3x ``` This operation defines a new function called the sum of f and g, denoted f + g.

Question 39

How do you subtract two functions?

Answer 39

f(x) - g(x) same shit

Question 40

How about dividing and multiplying?

Answer 40

Same shit

Question 41

How do we decide the domain of the new function?

Answer 41

For each of the four new functions, the domain is the set of all x that belong to both the domain of f and the domain of g, with the domain of the quotient further restricted to exclude any value of x for which g(x) = 0.

Question 42

A special case of fg deserves separate mention. how do we define cf by for any real number c and any function f?

Answer 42

(cf)(x) = c.f(x)

Question 43

What is this restricted case of product called?

Answer 43

The scalar product

Question 44

How else can you combine functions?

Answer 44

We can also combine two functions by first applying one function to an input and then applying the other function to the output of the first.

Question 45

for functions f : x -> y and g : y -> z what is the composite of f with g?

Answer 45

for functions f : x -> y and g : y -> z the composite of f with g is the function f ° g : X -> Z defined by (f ° g)(x) = f(g(x)) where the domain of f ° g is the set of all those x in the domain of g such that g.x/ is in the domain of f.

Question 46

How is the function f^-1 read?

Answer 46

f inverse and called the inverse of f.

Question 47

What is meant by a one to one function?

Answer 47

A function f that satisfies for all a and b; if f(a) = f(b) then a = b or for all a and b; if a /= b then f(a) /= f(b)

Question 48

is f(x) = x^2 one to one?

Answer 48

No: f(-1) = -1^2 = 1 = (1)^2 = 1 and -1 /= 1 shows that the squaring function is not one-to-one

Question 49

How do the domains and ranges of f compare to that of f^-1? Are they the same?

Answer 49

In general the domains and ranges of f and f^-1 are the same

Question 50

f^-1(f(x)) = ?

Answer 50

f^-1(f(x)) = x for all x in the domain of f f(f^-1(y)) = y

Question 51

Is it common for a linear equation to be one to one?

Answer 51

A linear function is always one to one. This can be proven with the formulas

Question 52

If f and g are one-to-one functions, what does this mean for their composite? (2)

Answer 52

The compositye f °g is also one to one and (f °g)^-1 = f ^-1°g^-1

Question 53

Many equations take the form f(x) = 0, where f is a function. If f is a one-to-one function, what is the solution?

Answer 53

Applying f ^-1 to both sides of f(x) = 0 gives f ^-1. (f(x)) = f ^-1(0), and (f(x)) = x shows that x = f ^-1(0) is the only possible solution. Since f(f ^-1(0)) = 0, f ^-1(0) is indeed a solution.

Question 54

It may happen that a function f whose domain is the natural one, consisting of all elements for which the defining rule makes sense, is not one-to-one. How may this sometimes be made one to one?

Answer 54

It may happen that a function f whose domain is the natural one, consisting of all ele- ments for which the defining rule makes sense, is not one-to-one, and yet a one-to-one function g can be obtained by restricting the domain of f For example, we have shown that the function f(x) = x^2 is not one-to-one but the function g(x) = x^2 with domain explicitly given as (0, ∞) is one-to-one.

Question 55

What does a rectangular coordinate system allow for?

Answer 55

A rectangular coordinate system allows us to specify and locate points in a plane. It also provides a geometric way to graph equations in two variables, in particular those arising from functions.

Question 56

What is meant by the origin of a coordinate system?

Answer 56

In a plane, two real-number lines, called coordinate axes, are constructed per- pendicular to each other so that their origins coincide. Their point of intersection is called the origin of the coordinate system.

Question 57

The coordinate axes divide the plane into four regions. What are these four regions called?

Answer 57

Quadrants

Question 58

How would we find x and y coordinates from y = x^2 + 2x - 3 ?

Answer 58

y = x^2 + 2x - 3 y = (0)^2 + 2(0) - 3 y = - 3 therefore (0, -3) is a point in the function same can be found by subbing values for y to find x

Question 59

What does the graph of a function display?

Answer 59

The graph of a function is the geometric representation of all its solutions. Since the equation has infinitely many solutions, it seems impossible to determine its graph precisely. However, we are concerned only with the graph’s general shape. For this reason, we plot enough points so that we can intelligently guess its proper shape.

Question 60

How would you determine the intercepts of the graph x = 3?

Answer 60

We can think of x = 3 as an equation in the variables x and y if we write it as x = 3 + 0y .Here y can be any value, but x must be 3. Because x = 3 when y = 0,the x intercept is (3, 0). There is no y-intercept, because x cannot be 0. The graph is a vertical line.

Question 61

Each function f gives rise to an equation, namely y = f (x), which is a special case of the equations we have been graphing. What does the graph consist of and what is the axis called?

Answer 61

Its graph consists of all points (x , f(x)), where x is in the domain of f. The vertical axis can be labeled either y or f(x), where f is the name of the function, and is referred to as the function-value axis.

Question 62

What is the basis of the vertical line test?

Answer 62

A useful geometric observation is that the graph of a function has at most one point of intersection with any vertical line in the plane. Recall that the equation of a vertical line is necessarily of the form x = a, where a is a constant Conversely, if a set of points in the plane has the property that any vertical line intersects the set at most once, then the set of points is actually the graph of a function. (The domain of the function is the set of all real numbers a with the property that the line x = a does intersect the given set of points, and for such an a the corresponding function value is the y-coordinate of the unique point of intersection of the line x = a and the given set of points.)

Question 63

How would you graph f: (-∞, ∞) -> (-∞, ∞) given by f (x) = sqrt(x)? ?

Answer 63

Recall that px denotes the principal square root of x. Thus, f(9) = sqrt(9) = 3, not +/-3. Also, the domain of f is [0, ∞) because its values are declared to be real numbers. Let us now consider intercepts. If f (x) = 0, then sqrt(x) = 0, so that x = 0. Also, if x = 0, then f (x) = 0. Thus, the x-intercept and the vertical-axis intercept are the same, namely, (0,0).

Question 64

How would the graph of p = G(g) = |p| look?

Answer 64

Notice that the q- and p-intercepts are the same point, (0,0). The graph would form a perfect correlation in the positive plane and one mirroring it in the quadrant to the left, converging at (0,0)

Question 65

There is an easy way to tell whether a curve is the graph of a function. Describe this way

Answer 65

If with the given x there are associated two values of y: y1 and y2, the curve is not the graph of a function of x. Looking at it another way, we have the following general rule, called the vertical-line test. If a vertical line, L, can be drawn that intersects a curve in at least two points, then the curve is not the graph of a function of x. When no such vertical line can be drawn, the curve is the graph of a function of x.

Question 66

What is the horizontal line function and what does it test for?

Answer 66

After we have determined whether a curve is the graph of a function, perhaps using the vertical-line test, there is an easy way to tell whether the function in question is one- to-one. If distinct input values (e.g -4 and 4) produce the same output, the function is not one-to-one. Looking at it another way, we have the following general rule, called the horizontal- line test. If a horizontal line, L, can be drawn that intersects the graph of a function in at least two points, then the function is not one-to-one. When no such horizontal line can be drawn, the function is one-to-one

Question 67

How do you measure the steepness of a line and what is this called

Answer 67

rise / run: y1 - y2 / x1 - x2; slope

Question 68

What is the slop of a vertical line?

Answer 68

A vertical line does not have a slope, because any two points on it must have x1 = x2, which gives a denominator of zero in Equation. For a horizontal line, any two points must have y1 = y2, This gives a numerator of zero, and hence the slope of the line is zero.

Question 69

How can we characterise the orientation of a line by its slope?

Answer 69

Zero slope: Undefined slope: Positive slope: Negative slope: horizontal line: line rises from left to right line vertical line: falls from left to right

Question 70

How do we find the equation of a line given a slope and a point on the line?

Answer 70

Suppose that line L has slope m and passes through the point (x1, y1). If (x, y) is any other point on L , we can find an algebraic relationship between x and y. Using the slope formula on the points (x1, y1) and (x, y) gives y1 - y2 / x1 - x2 = m y1 - y2 = m(x1 - x2) Every point on L satisfies this equation. It is also true that every point satisfying this equation must lie on L.

Question 71

Thus, this equation ( y1 - y2 = m(x1 - x2)) is an equation for L and is given a special name. What is this name?

Answer 71

y1 - y2 = m(x1 - x2) is a point-slope form of an equation of the line through (x1 , y1) with slope m

Question 72

How can we determine the equation of a line from two points?

Answer 72

First solve to get the slope then use this to plug the first coordinates into the point-slope form of an equation

Question 73

What is the slope intercept form of an equation?

Answer 73

Recall that point (b, 0) is the y intercept. If the slope m and y-intercept b of a line are known, an equation for the line is y - b = m(x - 0) solving for y gives: y = mx + b Which is the slope-intercept form of an equation of the line with slope m and y-intercept b.

Question 74

How would you find the slope and y intercept of a line with equation y = 5(3 - 2x)

Answer 74

``` y = 5(3 - 2x) y = 15 - 10x y = -10x + 15 ``` accoriding to y = mx + b, the slope is -10 and the y intercept is 15

Question 75

What is an equation of the vertical line through (-2, 3)?

Answer 75

x = -2

Question 76

What is an equation of the horizontal line through (-2, 3)?

Answer 76

y = 3

Question 77

What is meant. by a general linear equation?

Answer 77

From our discussions, we can show that every straight line is the graph of an equa- tion of the form Ax + By + C = 0, where A, B, and C are constants and A and B are not both zero. We call this a general linear equation (or an equation of the first degree)

Question 78

How are the variables x and y in the general linear equation said to be related?

Answer 78

x and y are said to be linearly related

Question 79

Give names for each of these forms of equations of straight lines ``` y = b y = mx + b x = a Ax + By + C = 0 y - y1 = x(x - x1) ```

Answer 79

``` y = b: Horizontal line y = mx + b: Slope-intercept form x = a: Vertical line Ax + By + C = 0: General linear form y - y1 = x(x - x1): Point-slope form ```

Question 80

When are two lines on a graph parallel?

Answer 80

Two lines are parallel if and only if they have the same slope or are both vertical.

Question 81

When are two lines on a graph perpendicular?

Answer 81

Two lines with slopes m1 and m2, respectively, are perpendicular to each other if and only if m1 = - 1/m2

Question 82

What is a quadratic function?

Answer 82

A function f is a quadratic function if and only if f.x/ can be written in the form f(x) = ax^2 + bx + c where a, b, and c are constants and a /= 0

Question 83

is g(x) = 1/x^2 quadratic?

Answer 83

no, because it cannot be written in the quadratic form

Question 84

What is the graph of a quadratic function called?

Answer 84

Parabola

Question 85

Describe the shape of a parabola and which variable(s) affect the direction

Answer 85

If a > 0, the graph extends upward indefinitely in a big U shape, and we say that the parabola opens upward. If a < 0, the parabola opens downward.

Question 86

what is meant by the axis of symmetry of a parabola?

Answer 86

Each parabola is symmetric about a vertical line, called the axis of symmetry of the parabola. That is, if the page were folded on one of these lines, then the two halves of the corresponding parabola would coincide. The axis (of symmetry) is not part of the parabola, but is a useful aid in sketching the parabola.

Question 87

what is meant by the vertex of a parabola?

Answer 87

The vertex is where the axis of symmetry cuts the parabola (the peak).

Question 88

What can be derived from the vertex of a parabola?

Answer 88

If a > 0, the vertex is the “lowest” point on the parabola. This means that f(x) has a minimum value at this point. By performing algebraic manipulations on ax2 + bx + c (referred to as completing the square), we can determine not only this minimum value, but also where it occurs.

Question 89

How do you determine the minimum value of a quadratic function?

Answer 89

f(x) = ax^2 + bx + c = (ax^2 + bx) + c adding and subtracting b^2 / 4a gives: (ax^2 + bx + b^2 / 4a ) + c - b^2 / 4a = a( x^2 + b/a x + b^2 / 4a^2) + c - b^2 / 4a so that: f(x) = a ( x + b/2a)^2 + c - b^2/4a since ( x + b/2a)^2 >= 0 and a > 0, it follows that f (x) has a minimum value when x + b/2a = 0, i.e x = - b/2a. The y coordinate corresponding to this value of x is f(- b/2a). Thus the vertex is given by: Vertex = ( - b/2a, f( - b/2a)) This is also the vertex of a parabola that opens downward .a < 0/, but in this case f( - b/2a) is the maximum value of f(x)

Question 90

Are quadratic functions one to one?

Answer 90

Observe that a function whose graph is a parabola is not one-to-one, in either the opening upward or opening downward case, since many horizontal lines will cut the graph twice

Question 91

How can you make a one to one function out of a quadratic function?

Answer 91

if we restrict the domain of a quadratic function to either [-b/21, ∞) or (-∞, -b/21], then the restricted function will pass the horizontal line test and therefore be one to one.

Question 92

How do we get the y intercept of a quadratic function?

Answer 92

The point where the parabola y = ax^2 + bx + c intersects the y-axis (that is, the y-intercept) occurs when x = 0. The y-coordinate of this point is c, so the y-intercept is c.

Question 93

How do you sketch a parabola from a quadrtatic function?

Answer 93

We can quickly sketch the graph of a quadratic function by first locating the vertex, the y-intercept, and a few other points, such as those where the parabola intersects the x-axis.

Question 94

What if there are no x intercepts?

Answer 94

In the event that the x-intercepts are very close to the vertex or that no x-intercepts exist, we find a point on each side of the vertex, so that we can give a reasonable sketch of the parabola.

Question 95

How would you graph 2q^2?

Answer 95

Here p is a quadratic function of q, where a = 2, b = 0, and c = 0. Since a > 0, the parabola opens upward and, thus, has a lowest point. The q-coordinate of the vertex is 0. Consequently, the minimum value of p is 0 and the vertex is (0, 0). In this case, the p-axis is the axis of symmetry. A parabola opening upward with vertex at (0, 0) cannot have any other intercepts. Hence, to draw a reason- able graph, we plot a point on each side of the vertex. If q = 2, then p = 8. This gives the point (2, 8) and, by symmetry, the point (-2, 8).