Ch1 - Algebraic and Non-Algebraic Functions and Their Inverse
Ch1 - Algebraic and Non-Algebraic Functions and Their Inverse
Ch1 - Algebraic and Non-Algebraic Functions and Their Inverse
LINES
The equation 𝑦 = 𝑚𝑥 + 𝑏 is called the slope−intercept equation of the line
with slope m and y−intercept b.
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:
𝑦
𝟐
(𝒙 − 𝒉) + (𝒚 − 𝒌) = 𝒂 𝟐 𝟐 𝑃(𝑥, 𝑦)
𝑦 𝑦 𝑥
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)2 = 𝑎2 shows that:
h = 1 , k = −5 and a = √3
(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)2 = 25
(𝑥 + 2)2 + (𝑦 − 3)2 = 4 , So c is (−2, 3)
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
PARABOLAS
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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
−
−
−
−
−
−
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
𝑦
2
𝑥 = −𝑦 𝑥 = 𝑦2
(a) (b)
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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
ELLIPSES
𝑥2 𝑦2
To sketch the graph of the equation: + = 1, compute a few values and
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plot the corresponding points, as shown in Fig.(1.4). The graph suggested by these
𝑥2 𝑦2
points belongs to a family of curves of the form ( 2
+ = 1) called ellipses.
𝑎 𝑏2
𝑥2 𝑦2
Fig.(1.4): The graph of + = 1.
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𝑥2 𝑦2
Now to graph + = 1:
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𝑥2 𝑥2 𝑦2
Find 𝑥 −intercept (by putting 𝑦 = 0), and since ≤ + = 1, it follows
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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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
HYPERBOLAS
𝑥2 𝑦2
Consider the graph of the equation: − = 1. Some of the points on this
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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.
𝑥2 𝑦2
Fig.(1.5): The graph of − = 1.
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𝑥2 𝑦2
Now to graph − = 1:
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𝑥2 𝑦2
Since = 1 + ≥ 1, it follows that 𝑥 2 ≥ 9, and therefore, |𝑥| ≥3. Hence,
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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
𝑥2 𝑦2 𝑏 𝑏
the hyperbola − = 1 are the lines 𝑦 = 𝑎 𝑥 and 𝑦 = − 𝑎 𝑥.
𝑎2 𝑏2
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).
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Fig.(1.6): Conic sections.
Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic 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
2
its own square root, 𝑦 = (√𝑦) for 𝑦 ≥ 0.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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:
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
Example (7): The following are some examples of equations that are functions.
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) 𝑥 = 𝑦2 If 𝑥 = 4, then 𝑦 = 2 or 𝑦 = −2.
(b) 𝑥 = |𝑦 + 3| If 𝑥 = 2, then 𝑦 = −5 or 𝑦 = −1.
(c) 𝑥 = −5 If 𝑥 = −5, then 𝑦 can be any real number.
(d) 𝑥 2 + 𝑦2 = 25 If 𝑥 = 0, then 𝑦 = 5 or 𝑦 = −5.
(e) 𝑦 = ±√𝑥 + 4 If 𝑥 = 5, then 𝑦 = +3 or 𝑦 = −3.
(f) 𝑥2 − 𝑦2 = 9 If 𝑥 = −5, then 𝑦 = 4 or 𝑦 = −4.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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.
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, ∞].
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
Example (10): Graph the following function over the interval [0, 1]
−𝑥, 𝑥<0
𝑓(𝑥) = { 𝑥 2 , 0≤𝑥≤1
1, 𝑥>1
Solution:
Homework (1):
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic 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
pass through the origin. Constant functions result when the slope 𝑚 = 0 (see
Fig.(1.12)).
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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.
(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).
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
𝟏 𝟏 𝟑 𝟐
(c) 𝒂 = , , , and
𝟐 𝟑 𝟐 𝟑
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 ) .)
𝟏 𝟏 𝟑 𝟐
Fig.(1.15): Graphs of 𝑓(𝑥) = 𝑥 𝑎 , 𝑓𝑜𝑟 𝒂 = , , , and .
𝟐 𝟑 𝟐 𝟑
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
Polynomials
A function p is a polynomial if:
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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
𝒚
𝑥 = 𝑦2
(𝒙, 𝒚)
(𝒙, −𝒚)
(a): Graphs of 𝑦 = 𝑥 2 and 𝑥 = 𝑦 2 (an even (b): Graph of 𝑦 = 𝑥 3 (an odd function)
functions) are symmetric about the y is symmetric about the origin.
and x−axis, respectively.
Example (11): Test whether the following functions are even, odd, or neither?
𝒇(𝒙) = 𝒙𝟐
Even function: (−𝑥)2 = 𝑥 2 for all 𝑥; symmetry about y−axis.
𝒇(𝒙) = 𝒙𝟐 + 1
Even function: (−𝑥)2 + 1 = 𝑥 2 + 1 for all 𝑥; symmetry about y−axis.
(see Fig.1.20a).
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).
𝑠𝑖𝑛𝜃 1 𝑐𝑜𝑠𝜃 1 1
𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, 𝑡𝑎𝑛𝜃 = , 𝑐𝑜𝑡𝜃= = , 𝑠𝑒𝑐𝜃 = , 𝑐𝑠𝑐𝜃 =
𝑐𝑜𝑠𝜃 𝑡𝑎𝑛𝜃 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic 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
𝜃 1 − cos 𝜃 B
𝑆𝑖𝑛2 =
2 2
𝜃 1 + cos 𝜃 𝒂
𝑐𝑜𝑠 2 = 𝒄
2 2
A C
The relationship between the 𝒃
angles and sides of a triangle may
Fig.(1.23): Relations between sides
be expressed using the Law of
and angles of a triangle.
Sines or the Law of Cosines
(see Fig.1.23).
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
Laws of Sines:
𝑎 𝑏 𝑐
= =
𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵 𝑠𝑖𝑛𝐶
Laws of Cosines:
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐 cos 𝐴
𝑏 2 = 𝑎2 + 𝑐 2 − 2𝑎𝑐 cos 𝐵
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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 (−∞, ∞).
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).
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
Hyperbolic Functions
𝑒 𝑥 − 𝑒 −𝑥 1 2
(a) 𝑠𝑖𝑛ℎ 𝜃 = (d) 𝑐𝑠𝑐ℎ 𝜃 = =
2 𝑠𝑖𝑛ℎ 𝜃 𝑒 𝑥 − 𝑒 −𝑥
𝑒 𝑥 + 𝑒 −𝑥 1 2
(b) 𝑐𝑜𝑠ℎ 𝜃 = (e) 𝑠𝑒𝑐ℎ 𝜃 = =
2 𝑐𝑜𝑠ℎ 𝜃 𝑒 𝑥 + 𝑒 −𝑥
𝑠𝑖𝑛ℎ 𝜃 𝑒 𝑥 − 𝑒 −𝑥 𝑐𝑜𝑠ℎ 𝜃 𝑒 𝑥 + 𝑒 −𝑥
(c) 𝑡𝑎𝑛ℎ 𝜃 = = (f) 𝑐𝑜𝑡ℎ 𝜃 = =
𝑐𝑜𝑠ℎ 𝜃 𝑒 𝑥 + 𝑒 −𝑥 𝑠𝑖𝑛ℎ 𝜃 𝑒 𝑥 − 𝑒 −𝑥
𝑐𝑜𝑠ℎ2 𝜃 − 𝑠𝑖𝑛ℎ2 𝜃 = 1
1 − 𝑡𝑎𝑛ℎ2 𝜃 = 𝑠𝑒𝑐ℎ2 𝜃
𝑐𝑜𝑡ℎ2 𝜃 − 1 = 𝑐𝑠𝑐ℎ2 𝜃
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
1 + √1 − 𝜃2
(d) 𝑠𝑒𝑐ℎ−1 𝜃 = ln ( ), 0<𝜃≤1
𝜃
1 √1 + 𝜃2
(e) 𝑐𝑠𝑐ℎ−1 𝜃 = ln ( + |𝜃|
), 𝜃≠0
𝜃
1 𝜃+1
(f) 𝑐𝑜𝑡ℎ−1 𝜃 = ln ( ), |𝜃| > 1
2 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.
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
(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).
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
Homework (2):
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
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Mathematics I / 1st. Semester / Dr. Rafi’ M.S. Ch.1: Algebraic and Non-Algebraic Functions
(d) 𝒇(𝒙) = 𝒙𝟐 − 𝟏
(e) 𝒇(𝒙) = |𝒙 + 𝟏| + |𝒙 − 𝟑|
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