Ch1 - Algebraic And Non-algebraic Functions And Their Inverse 5h171o

CHAPTER

1

Algebraic and Non−Algebraic Functions and Their Inverse  Equations and their graphs  Lines, Circles, Parabolas; Ellipses, Hyperbolas, and Conic sections.  Functions; Domain, Range, and their graphs.  Algebraic functions  Piecewise-defined functions, Linear functions, Power functions, Polynomial functions, Increasing and decreasing function, Rational functions, and Even and Odd functions.  Non−Algebraic or Transcendental functions  Trigonometric functions, Inverse Trigonometric functions, Logarithmic functions, Exponential functions, Hyperbolic functions, and Inverse Hyperbolic functions.  Shifting a Graph of a Function.

Equations and Their Graphs The graph of an equation involving 𝑥 and 𝑦 as its only variables consists of all points {𝑥, 𝑦) satisfying the equation.  LINES The equation 𝑦 = 𝑚𝑥 + 𝑏 is called the slope−intercept equation of the line with slope m and y−intercept b.

Example (1): Find the slope and y−intercept of the line: 2𝑥 − 𝑦 = 3? The equation is equivalent to 𝑦 = 2𝑥 − 3, which is the slope−intercept equation of the line with slope 𝑚 = 2 and у−intercept 𝑏 = −3.

Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions

 CIRCLES A circle is the set of points in a plane whose distances (radius) from a fixed point in the plane is constant.  The general equation of a circle with center at (h, k) is:

𝒙−𝒉

𝟐

+ 𝒚−𝒌

𝟐

=𝒂

𝟐

𝑦

𝑃(𝑥, 𝑦)

 The equation of a circle with its center at the origin is

𝒙𝟐 + 𝒚𝟐 = 𝒂𝟐

𝑐(ℎ, 𝑘) 𝑥

𝑦

𝑦

𝑦

𝑃(𝑥, 𝑦) 𝑥

𝑥 𝑐(0,0)

𝑐(ℎ, 0) 𝑐(0, 𝑘) 𝑥

Example (2): (a) Find the center and radius of the circle: 𝑥 − 1 2 + 𝑦 + 5 2 = 3

Comparing with: 𝑥 − ℎ 2 + 𝑦 − 𝑘 h = 1 , k = −5 and a = 3

2

= 𝑎2 shows that:

(b) If the circle: 𝑥 2 + 𝑦 2 = 25 is shifted 2 units to the left and 3 units up, find its

new equation? 𝑥 − (−2 )2 + 𝑦 − 3 𝑥+2

2

+ 𝑦−3

2

2

= 25

= 4 , So c is (−2, 3)

(c) What is the graph of the equation: 𝑥 2 + 𝑦 2 − 2𝑥 + 4𝑦 − 4 = 0?

Completing the square shows that the given equation is equivalent to the equation 𝑥 − 1 2 + 𝑦 + 2 2 = 9. Hence, its graph is the circle with center (1, −2) and radius 3. 2


 PARABOLAS The general form of parabola equation is 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄. The graph of this equation when 𝑎 ≠ 0 is a parabola. The parabola opens upward if 𝑎 > 0 and downward if 𝑎 < 0. The axis is the line 𝑏 𝑥 = − … … … … … … … … … … … . . (1) 2𝑎 The vertex of the parabola is the point where the axis and parabola intersect. Its 𝑏 𝑏 x−coordinate is 𝑥 = − ; and its y−coordinate is found by substituting 𝑥 = − 2𝑎 2𝑎 in the parabola’s equation.

Besides determining the direction in which the parabola 𝑦 = 𝑎𝑥 2 opens, the number a is a scaling factor. The parabola widens as a approaches zero and narrows as 𝑎 becomes large.

Sketch the graph of 𝒚 = 𝒙𝟐 , see Fig.(1.1): 1. Make a table of 𝑥𝑦 pairs that satisfy the function (substitute few values of 𝑥 and calculate the associated values of 𝑦). 2. Plot the points of (x, y) appear in the table. 3. Draw a smooth curve through the plotted points. These points suggest a curve, which belongs to a family of curves called parabolas.

3


Fig.(1.1): The graph of 𝑦 = 𝑥 2 .

In Fig.(1.1), we note that:  The graph of 𝑦 = 𝑥 2 contains the origin (0, 0) with all its points lie above the 𝑥 axis,  When 𝑥 are positive and increasing, 𝑦 increases without bound. Hence, in the first quadrant, the graph moves up without bound as it moves right.  Since −𝑥 2 = 𝑥 2 , it follows that, if any point 𝑥, 𝑦 lies on the graph in the first quadrant, then the point −𝑥, 𝑦 also lies on the graph in the second quadrant. Thus, the graph is symmetric with respect to the 𝑦 axis. The 𝑦 axis is called the axis of symmetry of this parabola. Sketch the graph of 𝒚 = −𝒙𝟐 If 𝑥, 𝑦 is on the graph of the parabola 𝑦 = 𝑥 2 (shown in Fig.1.1), then 𝑥, −𝑦 is on the graph of 𝑦 = −𝑥 2 , and vice versa. Hence, the graph of 𝑦 = −𝑥 2 is the reflection in the 𝑥 axis of the graph 𝑦 = 𝑥 2 . The result is the parabola shown in Fig.(1.2).

− − − − − −

Fig.(1.2): The graph of 𝑦 = −𝑥 2 .

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Sketch the graphs of 𝒙 = 𝒚𝟐 and 𝒙 = −𝒚𝟐 1. The graph of 𝑥 = 𝑦 2 is obtained from the parabola 𝑦 = 𝑥 2 (shown in Fig.1.1) by exchanging 𝑥 and 𝑦. The resulting curve is a parabola with the 𝑥 axis as its axis of symmetry and its "nose" at the origin (see Fig.(1.3b)). 2. The graph of 𝑥 = −𝑦 2 is the reflection in the 𝑦 axis of the graph 𝑥 = 𝑦 2 . The result is the parabola shown in Fig.(1.3a). 𝑦

𝑥 = −𝑦

2

𝑥 = 𝑦2

𝑥

(b)

(a)

Fig.(1.3): The graphs of 𝑥 = 𝑦 2 and 𝑥 = −𝑦 2 . 1

Example (3): Graph the equation 𝑦 = − 2 𝑥 2 − 𝑥 + 4? 1

Comparing the equation with 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 shows that 𝑎 = − , 𝑏 = −1, 2 𝑐 = 4. Since 𝑎 < 0, the parabola opens downward. From Equation (1) the axis is 𝑏 (−1) the vertical line 𝑥 = − = − = −1 2𝑎

2(−1/2)

When 𝑥 = −1, we have 1 9 𝑦 = − (−1)2 − −1 + 4 =

∴

2

2

The vertex is (−1, 9/2) The x−intercepts (put 𝑦 = 0): 1 − 𝑥2 − 𝑥 + 4 = 0 2 𝑥 2 + 2𝑥 − 8 = 0 𝑥−2 𝑥+4 =0 𝑥 = 2, 𝑥 = −4 Plot some points, and sketch the axis, complete the graph shown in Figure. 5


 ELLIPSES

𝑥2

𝑦2

To sketch the graph of the equation: + = 1, compute a few values and 9 4 plot the corresponding points, as shown in Fig.(1.4). The graph suggested by these points belongs to a family of curves of the form (

Fig.(1.4): The graph of

Now to graph

𝑥2 9

+

𝑦2 4

𝑥2 9

+

𝑥2

𝑦2

𝑎

𝑏2

+ 2

𝑦2 4

= 1) called ellipses.

= 1.

= 1: 𝑥2

𝑥2

𝑦2

 Find 𝑥 −intercept (by putting 𝑦 = 0), and since ≤ + = 1, it follows 9 9 4 that 𝑥 2 ≤ 9 , and therefore, 𝑥 = −3 and 𝑥 = 3. Its rightmost point is (3, 0), and its leftmost point is (−3, 0).  Find 𝑦 −intercept (by putting 𝑥 = 0) gives 𝑦 = −2 and 𝑦 = 2, and that its lowest point is (0, −2) and its highest point is (0, 2). In the first quadrant, as 𝑥 increases from 0 to 3, 𝑦 decreases from 2 to 0.  If (𝑥, 𝑦) is any point on the graph, then (−𝑥, 𝑦) also is on the graph. Hence, the graph is symmetric with respect to the 𝑦 axis. Similarly, if (𝑥, 𝑦) is on the graph, so is (𝑥, −𝑦), and therefore the graph is symmetric with respect to the 𝑥 axis. 𝑥2

𝑦2

When a = b, the ellipse 2 + 2 = 1 is a circle 𝑎 𝑏 with the equation 𝑥 2 + 𝑦 2 = 𝑎2 , that is, a circle with center at the origin and radius a. Thus, circles are special cases of ellipses. The standard equation of an ellipse with center at (h, k) is

𝑥−ℎ 𝑎2

2

𝑦−𝑘 + 𝑏2

2

Graph of the ellipse

=1

𝑥2

𝑦2

𝑎

𝑏2

+ 2

= 1, a > b,

where the major axis is horizontal.

6


 HYPERBOLAS

𝑥2

𝑦2

Consider the graph of the equation: − = 1. Some of the points on this 9 4 graph are tabulated and plotted in Fig.(1.5). These points suggest the curve shown 𝑥2

𝑦2

in the figure, which belongs to a family of curves of the form ( 2 − 2 = 1) called 𝑎 𝑏 hyperbolas.

Fig.(1.5): The graph of

Now to graph 𝑥2

𝑥2

−

𝑦2

𝑥2 9

−

𝑦2 4

= 1.

= 1:

9 𝑦2

4

𝑥2

𝑦2

 Since = 1 + ≥ 1, it follows that 𝑥 2 ≥ 9, and therefore, 𝑥 ≥3. Hence, 9 4 there are no points on the graph between the vertical lines 𝑥 = −3 and 𝑥 = 3.  If (𝑥, 𝑦) is on the graph, so is (−𝑥, 𝑦); thus, the graph is symmetric with respect to the 𝑦 axis. Similarly, the graph is symmetric with respect to the 𝑥 axis. 2 2  Note in Fig.(1.5); the dashed lines (𝑦 = 𝑥 and, 𝑦 = − 𝑥) are called the 3 3 asymptotes of the hyperbola: Points on the hyperbola get closer and closer to these asymptotes as they recede from the origin. In general, the asymptotes of the hyperbola

𝑎2

−

𝑏2

𝑏

𝑏

= 1 are the lines 𝑦 = 𝑎 𝑥 and 𝑦 = − 𝑎 𝑥.

 CONIC SECTIONS Parabolas, ellipses, and hyperbolas together make up a class of curves called conic sections. They can be defined geometrically as the intersections of planes with the surface of a right circular cone, as shown in Fig.(1.6). 7

Fig.(1.6): Conic sections.


Functions and Their Graphs  Functions; Domain and Range A function is defined as a set of ordered pairs (x, y), such that for each value of the independent variable x, there corresponds only one value of the dependent variable y. The set of x values is called the domain of the function, while the set of all corresponding values of 𝑦 is called the range of the function. The notation f (x) is often used in place of y to indicate the value of the function f for a specific x and is read “f of x” or “f at x.”

Example (4): the domains and ranges of these functions.

Solution:

 The formula 𝑦 = 𝑥 2 gives a real y−value for any real number x, so the domain is(−∞, ∞). The range of 𝑦 = 𝑥 2 is [0, ∞] because the square of any real number is nonnegative and every nonnegative number y is the square of its own square root, 𝑦 =

𝑦

2

for 𝑦 ≥ 0.

 The formula 𝑦 = 1/𝑥 gives a real y−value for every x except 𝑥 = 0. we cannot divide any number by zero. The range of 𝑦 = 1/𝑥, the set of reciprocals of all nonzero real numbers, is the set of all nonzero real numbers, since 𝑦 = 1/(1/𝑦).  The formula 𝑦 = 𝑥 gives a real y−value only if 𝑥 ≥ 0. The range of 𝑦 = 𝑥 is [0, ∞] because every nonnegative number is some number’s square root (namely, it is the square root of its own square). 8


 In 𝑦 = 4 − 𝑥, the quantity 4 − 𝑥 cannot be negative. That is, 4 − 𝑥 ≥ 0, or 𝑥 ≤ 4. The formula gives real 𝑦 −values for all 𝑥 ≤ 4. The range of 4 − 𝑥 is [0, ∞], the set of all nonnegative numbers.  The formula 𝑦 = 1 − 𝑥 2 gives a real y−value for every x in the closed interval from −1 to 1. Outside this domain, 1 − 𝑥 2 is negative and its square root is not a real number. The values of 1 − 𝑥 2 vary from 0 to 1 on the given domain, and the square roots of these values do the same. The range of 1 − 𝑥 2 is [0, 1].

 Graphs of Functions The graph of a function 𝑓 is the graph of the equation: 𝑦 = 𝑓(𝑥). If (x, y) is a point on the graph, then 𝑦 = 𝑓(𝑥) is the height of the graph above the point x if 𝑓(𝑥) is positive or below x if 𝑓(𝑥) is negative (see Fig.(1.7)). Fig.(1.7): If (x, y) lies on the graph of f, then the value 𝑦 = 𝑓(𝑥) is the height of the graph above the point x (or below x if ƒ(x) is negative)..

Example (5): Graph the function 𝑓 𝑥 = 𝑥 + 2 and find its domain and range. Solution:

The graph of 𝑓 is the graph of the equation 𝑦 = 𝑥 + 2, which is the straight line with slope 1 and 𝑦 intercept 2. The set of all real numbers is both the domain and range of 𝑓, (see Fig.(1.8)). Fig.(1.8): The graph of 𝑓 𝑥 = 𝑥 + 2 is the set of points (x, y) for which y has the value x + 2.

9


Example (6): Graph the function over the interval [−2, 2] Solution: 1. Make a table of 𝑥𝑦 −pairs that satisfy the function, in this case: y = 𝑥 2 . 2. Plot the points (x, y) whose coordinates appear in the table. 3. Draw a smooth curve through the plotted points, and label the curve with

its equation.

Example (7): The following are some examples of equations that are functions. (a) (b) (c) (d)

𝑦 = 3𝑥 + 1 𝑦 = 𝑥2 𝑦 = 𝑥 −5 𝑦 = −3

(e)

𝑦=

𝑥−3

(f)

𝑦 = 2𝑥 + 9

(g)

𝑦=

(h) (i)

𝑦 = tan 𝑥 𝑦 = cos 2𝑥

6

𝑥

𝑥 2 +4

Example (8): The following are some examples of equations that are not functions; each has an example to illustrate why it is not a function. (a) (b) (c) (d) (e) (f)

𝑥 = 𝑦2 𝑥 = 𝑦+3 𝑥 = −5 𝑥 2 + 𝑦2 = 25 𝑦 =± 𝑥+4 𝑥2 − 𝑦2 = 9

If If If If If If

𝑥 = 4, 𝑥 = 2, 𝑥 = −5, 𝑥 = 0, 𝑥 = 5, 𝑥 = −5,

then 𝑦 = 2 or 𝑦 = −2. then 𝑦 = −5 or 𝑦 = −1. then 𝑦 can be any real number. then 𝑦 = 5 or 𝑦 = −5. then 𝑦 = +3 or 𝑦 = −3. then 𝑦 = 4 or 𝑦 = −4. 10


Algebraic Functions An algebraic function is a function constructed from polynomials using algebraic operations (addition, subtraction, multiplication, division, and taking roots). Fig.(1.9) displays the graphs of three algebraic functions.

(a)

(b)

(c)

Fig.(1.9): Graphs of three algebraic functions.

 Piecewise-Defined Functions These functions are described by using different parts of its domain, such as the absolute value function.

Example (9): Graph the function 𝑥 and find its domain and range. The graph of 𝑓 𝑥 = 𝑦 = 𝑥 is shown in Fig.(1.9). Notice that 𝑓 𝑥 = 𝑥 when 𝑥 ≥ 0, whereas, 𝑓 𝑥 = −𝑥 when 𝑥 ≤ 0. the domain of 𝑓 consists of all real numbers (−∞, ∞) , but the range is the set of all nonnegative real numbers [0, ∞].

Fig.(1.9): The absolute value function has domain (−∞, ∞) and range [0, ∞].

11


Example (10): Graph the following function over the interval [0, 1] −𝑥, 𝑥2, 𝑓 𝑥 = 1,

Solution:

𝑥<0 0≤𝑥≤1 𝑥>1

The values of ƒ are given by; 𝑦 = −𝑥 when 𝑥 < 0, 𝑦 = 𝑥 2 when 0 ≤ 𝑥 ≤ 1, and 𝑦 = 1 when 𝑥 > 1. The function, however, is just one function whose domain is the entire set of real numbers (see Fig.1.10).

Fig.(1.10): To graph the function 𝑦 = 𝑓(𝑥) shown here, we apply different formulas to different parts of its domain.

Homework (1): 

Find the domain and range of each of the following functions:

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

Graph the following piecewise−defined functions: 1. 2. 3. 4. 5.

 Linear Functions A function of the form 𝑓 𝑥 = 𝑚𝑥 + 𝑏, for constants m and b, is called a linear function. Fig.(1.11) shows an array of lines 𝑓 𝑥 = 𝑚𝑥 where 𝑏 = 0, so these lines through the origin. Constant functions result when the slope 𝑚 = 0 (see Fig.(1.12)).

Fig.(1.11): The collection of 𝑦 = 𝑚𝑥 lines has slope m and all lines through the origin.

13

Fig.(1.12): A constant function has slope 𝑚 = 0.


 Power Functions A function 𝑓 𝑥 = 𝑥 𝑎 where a is a constant, is called a power function. There are several important cases to consider. (a)

𝒂 = 𝒏, a positive integer.

The graphs of 𝑓 𝑥 = 𝑥 𝑛 , for 𝑛 =1, 2, 3, 4, 5, are displayed in Fig.(1.13). These functions are defined for all real values of x. Notice that as the power n gets larger, the curves tend to flatten toward the x−axis on the interval (−1, 1), and also rise more steeply for 𝑥 >1. Each curve es through the point (1, 1) and through the origin.

Fig.(1.13): Graphs of 𝑓 𝑥 = 𝑥 𝑛 , 𝑛 = 1, 2, 3, 4, 5 defined for −∞ < 𝑥 < ∞.

(b) 𝒂 = −𝟏

𝐨𝐫 𝒂 = −𝟐

The graphs of the functions 𝑓 𝑥 = 𝑥 −1 = 1/𝑥 and g 𝑥 = 𝑥 −2 = 1/𝑥 2 are shown in Fig.(1.14). Both functions are defined for all 𝑥 ≠ 0. The graph of 𝑦 = 1/𝑥 is the hyperbola 𝑥𝑦 = 1 which approaches the coordinate axes far from the origin, and the graph of y = 1/𝑥 2 also approaches the coordinate axes.

Fig.(1.14): Graphs of 𝑓 𝑥

for part (a) and for part (b). 14


(c) 𝒂 =

𝟏 𝟐

,

𝟏 𝟑

,

𝟑

𝟐

𝟐

𝟑

, and

3

The functions 𝑓 𝑥 = 𝑥 1/2 = 𝑥 and g 𝑥 = 𝑥 1/3 = 𝑥 are the square root and cube root functions, respectively. The domain of the square root function is [0, ∞], but the cube root function is defined for all real x. Their graphs are displayed in Fig.(1.15) along with the graphs of y = 𝑥 3/2 and y = 𝑥 2/3 . (Recall that 3 2 𝑥 3/2 = 𝑥 1/2 and 𝑥 2/3 = 𝑥 1/3 .)

Fig.(1.15): Graphs of 𝑓 𝑥 = 𝑥 𝑎 , 𝑓𝑜𝑟 𝒂 =

15

𝟏 𝟐

,

𝟏 𝟑

,

𝟑 𝟐

, and

𝟐 𝟑

.


 Polynomials A function p is a polynomial if: 𝑝 𝑥 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2 + ⋯ … … + 𝑎𝑛 −1 𝑥 𝑛−1 + 𝑎𝑛 𝑥 𝑛 where, n is a positive integer and the numbers 𝑎0, 𝑎1, 𝑎2, …….., 𝑎𝑛 are real constants (called the coefficients of the polynomial). All polynomials have domain (−∞, ∞). If the leading coefficient 𝑎𝑛 ≠ 0 and 𝑛 > 0, then n is called the degree of the polynomial. Linear functions; 𝑓 𝑥 = 𝑚𝑥 + 𝑏, with 𝑚 ≠ 0 are polynomials of degree 1. Quadratic functions are polynomials of degree 2 and written as, 𝑝 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. Likewise, cubic functions are polynomials of degree 3 and written as, 𝑝 𝑥 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑. Fig.(1.16) shows the graphs of three polynomials.

Fig.(1.16): Graphs of three polynomial functions.

 Increasing and Decreasing Functions If the graph of a function rises as you move from left to right, we say that the function is increasing, and if the graph falls as you move from left to right, the function is decreasing. Some examples of these functions are shown in Fig.(1.17).

16


Fig.(1.17): Graphs of increasing and / or decreasing functions.

17


 Rational Functions A rational function is a quotient or ratio of two polynomials: 𝑝(𝑥) 𝑓 𝑥 = 𝑞(𝑥) where, p and q are polynomials. The domain of a rational function is the set of all real x for which, 𝑞(𝑥) ≠ 0. For example, the function

2𝑥 2 − 3 𝑓 𝑥 = 7𝑥 + 4

is a rational function with domain 𝑥 𝑥 ≠ −4/7 . Its graph is shown in Fig.(1.18a) with graphs of two other rational functions in Figs.(1.18b and 1.18c).

Fig.(1.18): Graphs of three rational functions.

 Even Functions and Odd Functions

18

Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions 𝒚

𝑥 = 𝑦2 (𝒙, 𝒚)

𝒙

(𝒙, −𝒚)

(a): Graphs of 𝑦 = 𝑥 2 and 𝑥 = 𝑦 2 (an even functions) are symmetric about the y and x−axis, respectively.

(b): Graph of 𝑦 = 𝑥 3 (an odd function) is symmetric about the origin.

Fig.(1.19): Graphs of even and odd functions.

Example (11): Test whether the following functions are even, odd, or neither?  𝒇 𝒙 = 𝒙𝟐

Even function: −𝑥 𝟐  𝒇 𝒙 =𝒙 +1 Even function: −𝑥 (see Fig.1.20a).

2

= 𝑥 2 for all 𝑥; symmetry about y−axis.

2

+ 1 = 𝑥 2 + 1 for all 𝑥; symmetry about y−axis.

 𝒇 𝒙 =𝒙

Odd function: −𝑥 = −𝑥 for all 𝑥; symmetry about the origin.  𝒇 𝒙 =𝒙+1 Not odd: 𝑓 −𝑥 = −𝑥 + 1 , but −𝑓 𝑥 = −𝑥 − 1. The two are not equal. Not even: −𝑥 + 1 ≠ 𝑥 + 1 for all 𝑥 ≠ 0 (see Fig.1.20b).

Fig.(1.20): Even, odd and neither functions for Example (9).

19


Non-Algebraic or Transcendental Functions  Trigonometric Functions The six basic trigonometric functions are: 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, 𝑡𝑎𝑛𝜃 =

𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃

, 𝑐𝑜𝑡𝜃 =

1 𝑡𝑎𝑛𝜃

=

𝑐𝑜𝑠𝜃 𝑠𝑖𝑛𝜃

, 𝑠𝑒𝑐𝜃 =

1 𝑐𝑜𝑠𝜃

, 𝑐𝑠𝑐𝜃 =

1 𝑠𝑖𝑛𝜃

These functions are defined using a circle with equation 𝑥 2 + 𝑦 2 = 𝑟 2 and the angle 𝜃 in standard position as shown in Fig.(1.21) with its vertex at the center of the circle and its initial side along the positive portion of the x−axis.

Fig.(1.21): Defining of trigonometric

Functions in of 𝑥, 𝑦 and 𝑟.

Fig.(1.22): Angles in standard position in the xy−plane.

20


The variable 𝜃 is generally expressed in radians (π radians = 180o). For real values of 𝜃, 𝑠𝑖𝑛𝜃 and 𝑐𝑜𝑠𝜃 lie between −1 and 1 inclusive. The following are some properties of these functions: 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 1 + 𝑡𝑎𝑛2 𝜃 = 𝑠𝑒𝑐 2 𝜃 1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃 𝑠𝑖𝑛 −𝜃 = −𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 −𝜃 = 𝑐𝑜𝑠 𝜃 𝑡𝑎𝑛 −𝜃 = −𝑡𝑎𝑛 𝜃 𝑠𝑖𝑛 𝜃 + 2𝜋 = 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 + 2𝜋 = 𝑐𝑜𝑠 𝜃 𝑡𝑎𝑛 𝜃 + 𝜋 = 𝑡𝑎𝑛 𝜃 𝑠𝑖𝑛⁡ (𝐴 ± 𝐵) = 𝑠𝑖𝑛𝐴 𝑐𝑜𝑠𝐵 ± 𝑐𝑜𝑠𝐴 𝑠𝑖𝑛𝐵 𝑐𝑜𝑠 𝐴 ± 𝐵 = 𝑐𝑜𝑠𝐴 𝑐𝑜𝑠𝐵 ∓ 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑡𝑎𝑛 𝐴 ± 𝐵 =

𝑡𝑎𝑛𝐴 ± 𝑡𝑎𝑛𝐵 1 ∓ 𝑡𝑎𝑛𝐴 𝑡𝑎𝑛 𝐵

𝑠𝑖𝑛 2𝜃 = 2 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 2𝜃 = 𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃 = 2 𝑐𝑜𝑠 2 𝜃 − 1

= 1 − 2 𝑠𝑖𝑛2 𝜃

𝜃 1 − cos 𝜃 = 2 2 𝜃 1 + cos 𝜃 𝑐𝑜𝑠 2 = 2 2

B

𝑆𝑖𝑛2

𝒂

𝒄

The relationship between the angles and sides of a triangle may be expressed using the Law of Sines or the Law of Cosines (see Fig.1.23). 21

A

𝒃

Fig.(1.23): Relations between sides

and angles of a triangle.

C


Laws of Sines:

𝑎 𝑠𝑖𝑛𝐴

Laws of Cosines:

=

𝑏 𝑠𝑖𝑛𝐵

=

𝑐 𝑠𝑖𝑛𝐶

𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴 𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶

The graphs of the six trigonometric functions are shown in Fig.(1.24)

Fig.(1.24): Graphs of the (a) cosine, (b) sine, (c) tangent, (d) secant, (e) cosecant, and (f) cotangent functions using radian measure. The shading for each trigonometric function indicates its periodicity.

22


 Inverse Trigonometric Functions (a)

𝑦 = 𝑠𝑖𝑛−1 𝜃,

(b)

𝑦 = 𝑐𝑜𝑠 −1 𝜃, (0 ≤ 𝑦 ≤ 𝜋)

(c)

𝑦 = 𝑡𝑎𝑛−1 𝜃, (−𝜋/2 ≤ 𝑦 ≤ 𝜋/2)

(d)

𝑦 = 𝑐𝑠𝑐 −1 𝜃 = 𝑠𝑖𝑛−1 1/𝜃,

(e)

𝑦 = 𝑠𝑒𝑐 −1 𝜃 = 𝑐𝑜𝑠 −1 1/𝜃, (0 ≤ 𝑦 ≤ 𝜋)

(f)

𝑦 = 𝑐𝑜𝑡 −1 𝜃 = − 𝑡𝑎𝑛−1 𝜃, (0 ≤ 𝑦 ≤ 𝜋)

(−𝜋/2 ≤ 𝑦 ≤ 𝜋/2)

(−𝜋/2 ≤ 𝑦 ≤ 𝜋/2)

𝜋

2

 Logarithmic Functions These functions are of the form, 𝑓 𝑥 = log 𝑎 𝑥, where the base 𝑎 ≠ 0, 1 is a positive constant. These and the exponential functions are inverse functions. If a = e = 2.71828….called the natural base of logarithms, we write 𝑓 𝑥 = log 𝑒 𝑥 = ln 𝑥, called the natural logarithm of x. Fig.(1.25) shows the graphs of four logarithmic functions with various bases. In each case the domain is (0, ∞) and the range is −∞, ∞ .

Fig.(1.25): Graphs of four logarithmic functions.

 Exponential Functions These are functions of the form, 𝑓 𝑥 = 𝑎 𝑥 , where the base 𝑎 ≠ 0,1 is a positive constant. All exponential functions have domain (−∞, ∞) and range 0, ∞ . The graphs of some exponential functions are shown in Fig.(1.26). 23


Fig.(1.26): Graphs of exponential functions.

 Hyperbolic Functions (a)

𝑠𝑖𝑛ℎ 𝜃 =

(b)

𝑐𝑜𝑠ℎ 𝜃 =

(c)

𝑡𝑎𝑛ℎ 𝜃 =

𝑒 𝑥 − 𝑒 −𝑥

(d) 𝑐𝑠𝑐ℎ 𝜃 =

2 𝑒 𝑥 + 𝑒 −𝑥

(e) 𝑠𝑒𝑐ℎ 𝜃 =

2 𝑠𝑖𝑛 ℎ 𝜃 𝑐𝑜𝑠 ℎ 𝜃

=

𝑒 𝑥 − 𝑒 −𝑥

(f) 𝑐𝑜𝑡ℎ 𝜃 =

𝑒 𝑥 + 𝑒 −𝑥

The following are some properties of these functions: 𝑐𝑜𝑠ℎ2 𝜃 − 𝑠𝑖𝑛ℎ2 𝜃 = 1 1 − 𝑡𝑎𝑛ℎ2 𝜃 = 𝑠𝑒𝑐ℎ2 𝜃 𝑐𝑜𝑡ℎ2 𝜃 − 1 = 𝑐𝑠𝑐ℎ2 𝜃 𝑠𝑖𝑛ℎ −𝜃 = −𝑠𝑖𝑛ℎ 𝜃 𝑐𝑜𝑠ℎ −𝜃 = 𝑐𝑜𝑠ℎ 𝜃 𝑡𝑎𝑛ℎ −𝜃 = −𝑡𝑎𝑛ℎ 𝜃 𝑠𝑖𝑛ℎ⁡ (𝐴 ± 𝐵) = 𝑠𝑖𝑛ℎ𝐴 𝑐𝑜𝑠ℎ𝐵 ± 𝑐𝑜𝑠ℎ𝐴 𝑠𝑖𝑛ℎ𝐵 𝑐𝑜𝑠ℎ 𝐴 ± 𝐵 = 𝑐𝑜𝑠ℎ𝐴 𝑐𝑜𝑠ℎ𝐵 ± 𝑠𝑖𝑛ℎ𝐴 𝑠𝑖𝑛ℎ𝐵 𝑡𝑎𝑛ℎ 𝐴 ± 𝐵 =

𝑡𝑎𝑛ℎ 𝐴 ± 𝑡𝑎𝑛ℎ 𝐵 1 ± 𝑡𝑎𝑛ℎ 𝐴 𝑡𝑎𝑛ℎ 𝐵 24

1 𝑠𝑖𝑛 ℎ 𝜃 1 𝑐𝑜𝑠 ℎ 𝜃 𝑐𝑜𝑠 ℎ 𝜃 𝑠𝑖𝑛 ℎ 𝜃

= = =

2 𝑒 𝑥 − 𝑒 −𝑥 2 𝑒 𝑥 + 𝑒 −𝑥 𝑒 𝑥 + 𝑒 −𝑥 𝑒 𝑥 − 𝑒 −𝑥


 Inverse Hyperbolic Functions When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown below: (a)

𝑠𝑖𝑛ℎ−1 𝜃 = ln 𝜃 + 𝜃 2 + 1 , 𝑎𝑙𝑙 𝜃

(b)

𝑐𝑜𝑠ℎ−1 𝜃 = ln 𝜃 + 𝜃 2 + 1 , 𝜃 ≥ 1

(c)

𝑡𝑎𝑛ℎ−1 𝜃 =

(d)

𝑠𝑒𝑐ℎ−1 𝜃 = ln

1 + 1 − 𝜃2

(e)

𝑐𝑠𝑐ℎ−1 𝜃 = ln

1

(f)

𝑐𝑜𝑡ℎ−1 𝜃 =

1

ln

2

1 2

1+𝜃

,

1−𝜃 𝜃

𝜃

ln

+

1 + 𝜃2

𝜃 +1 𝜃 −1

𝜃

,

𝜃 <1 ,

0<𝜃≤1 ,

𝜃≠0 𝜃 >1

Example (12): Identify each function given here as one of the types of functions we have discussed. Keep in mind that some functions can fall into more than one category. For example, 𝑓 𝑥 = 𝑥 2 is both a power function and a polynomial of second degree.

25


Shifting a Graph of a Function  To shift the graph of a function 𝑦 = 𝑓(𝑥) straight up, add a positive constant to the right−hand side of 𝑦 = 𝑓 𝑥 .  To shift the graph of a function 𝑦 = 𝑓(𝑥) straight down, add a negative constant to the right−hand side of 𝑦 = 𝑓(𝑥).  To shift the graph of 𝑦 = 𝑓(𝑥) to the left, add a positive constant to x.  To shift the graph of 𝑦 = 𝑓(𝑥) to the right, add a negative constant to x. Shift Formulas  Vertical Shifts 𝑦 =𝑓 𝑥 +𝑘 Shifts the graph of f up 𝑘 units if k > 0 Shifts it down 𝑘 units if k < 0  Horizontal Shifts 𝑦 =𝑓 𝑥+ℎ Shifts the graph of f left ℎ units if h > 0 Shifts it right ℎ units if h < 0 EXAMPLE (13): Shifting a Graph (a) Adding 1 to the right-hand side of the formula 𝑦 = 𝑥 2 to get 𝑦 = 𝑥 2 + 1 shifts the graph up 1 unit (Fig.1.27). (b) Adding −2 to the right-hand side of the formula 𝑦 = 𝑥 2 to get 𝑦 = 𝑥 2 − 2 shifts the graph down 2 units (Fig.1.27).

Fig.(1.27): To shift 𝑦 = 𝑥 2 up (or down), we add positive (or negative) constants to the formula for f (Example 13a and b).

26


(c) Adding 3 to x in 𝑦 = 𝑥 2 to get 𝑦 = (𝑥 + 3)2 shifts the graph 3 units to the left (Fig.1.28).

Fig.(1.28): To shift the graph 𝑦 = 𝑥 2 to the left, we add a positive constant to 𝑥. To shift the graph to the right, we add a negative constant to 𝑥 (Example 13c).

(d) Adding −2 to x in 𝑦 = 𝑥 and then adding −1 to the result, gives 𝑦 = 𝑥 − 2 − 1 and shifts the graph 2 units to the right and 1 unit down (Fig.1.29).

Fig.(1.29): Shifting the graph 𝑦 = 𝑥 2 units to the right and 1 unit down (Example 13d).

27


Homework (2):

28


29


30


31


32


33


34


35


36


(d) 𝒇 𝒙 = 𝒙𝟐 − 𝟏 (e) 𝒇 𝒙 = 𝒙 + 𝟏 + 𝒙 − 𝟑

37

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