1.1 Functions
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Transcript of 1.1 Functions
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Chapter 1:
Functions
Chapter 2:
Limits
Chapter 3:
Continuity
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Objectives:
Work with functional notations and use them to
express the relation between variables.
Define the domain and determine the range of a
function.
Draw the graph of function.
Perform operations on functions
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Function -- is a rule that assigns to each element X in set A
exactly one element called f(x) in set B.
Four ways to represent functions
1. Verbally: The Circumference of a circle increases with its radius.
2. Numerically: Table Values
Quiz Score
1 95
2 98
3. Graphically
4. Algebraically (Use of explicit formula)
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HOW FUNCTIONS ARE DEFINED:
a. Explicitly: y = f (x)
y = x2 –x +2
b. Implicitly: f (x, y) = 0
x2 + xy2 = 9
c. Parametric form: x = f(t), y = g(t)
* t = parameter
x = 2t – 1 y = t2
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SYMMETRY:
The function is an EVEN function if f (x) = f (-x).
The graph of the function is symmetric with
respect to the vertical axis. (E.g. y = x2 )
The function is an ODD function if f (- x) = - f (x).
The graph of the function is symmetric with
respect to the origin. (E.g. y = x3 )
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CLASSIFICATION OF FUNCTIONS:
1. Algebraic functions
2. Transcendental functions
a. trigonometric
b. inverse trigonometric
c. exponential
d. logarithmic
e. hyperbolic
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“Domain and Range of a function”
Set A is the domain of the function, while x is the independent
variable.
Set B is the range of the function, while f (x) is the dependent
variable.
Domain: { x | -2 x 2 }
Range: { f (x) | 0 f (x) 2 }
*Restriction is necessary for f (x) to be real.
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Example 1: Find the natural domain and range of :
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Example 2: Find the natural domain and range of :
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Example 3: Find the natural domain and range of :
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Example 4: Find the natural domain of:
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Example 5: Find the natural domain and range of:
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Example 6: Find the natural domain of:
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“Piecewise Defined Function”
The function has different explicit formulas in different
intervals of its domain.
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Example 1: Draw the graph of:
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Example 2: Draw the graph of:
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Example 3: Draw the graph of:
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Example 4: Draw the graph of:
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Example 5: Draw the graph of:
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“Functions as Mathematical Models of Reality”
Mathematics can be used as a basis for decision
making. Many situations in real life can be
represented with a mathematical model, usually
as functions.
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Example 1: Let P be the perimeter of an equilateral
triangle. Write a formula A(P), the area of thetriangle as a function of the perimeter.
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Example 2: Express the area of a circle as a function
of its circumference.
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Example 3: Express the volume of a sphere as a
function of its (a) diameter (b) surface area
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Example 4: Express the surface area of a cube as a
function of its volume.
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Example 5: A rectangle has a perimeter of 20 inches.
Express the area as a function of one of itssides.
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Example 6: Write a formula describing the distance
of a point on the parabola x = 2y2 to (10,0) as afunction of x.
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Example 7: One of the legs of a right triangle has a
length of 4 cm. Express the length of thealtitude perpendicular to the hypotenuse as afunction of the length of the hypotenuse.
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Example 8: Boxes are to be made from rectangular
cardboards, 8 inches by 15 inches. Equalsquares are to be cut from the four corners,then the flaps are folded upward. Express thevolume of the box as a function of a side ofthe squares from the corners.