Review of Algebra and Trigonometry
Exponents and Radicals
For a nonzero real number a:
The square rootof a nonnegative real number b is real number a such that the square of a is equal to b.Theother even roots can be defined similarly. The odd roots are defined for all real numbers.
How many square roots does 4 have?
How many square roots does –4 have?
How many cube roots does –8 have?
For a real number a and any integer n greater than 1, we write:
.
Which of the following
is easier to calculate without a calculator?
Most common mistakes are arithmetic and simplification mistakes:
Examples: Try to find the right answer.
To find the value of an expression for x = 2 and y = -3:
It is better to substitute for x and y instead of simplifying the expression first.
Sometimes over simplification can cause problems:
If they want the equation of the line through (3,-2) with slope –4 the answer on a test is:
y – (-2) = -4( x – 3)
Most books ask you to simplify it and write it in the form: y = -4x + 10
Multiplication
and Factoring of Polynomials:
Rationalizing
denominators and numerators:
Rationalize the denominator:
Rationalize the numerator:
Verify the following:
Domain of a Function:
The set of real numbers x for which the function f(x) is defined is called domain of the function.
Function Domain
all real numbers
all real numbers
all real numbers x > -1.
Note: Justify the answers to examples on domains.
Evaluate f(-2), f(a), f(x+a).
Even and Odd Functions:
A function f is even if for every number x in its domain the number -x is also in the domain and
f(-x) = f(x)
A function f is odd if for every number x in its domain the number -x is also in the domain and
f(-x) = -f(x)
The graph of an even function is symmetric with respect to the y-axis and the graph of an odd function
is symmetric with respect to the origin.
Question: What is the importance of these concepts?
Identify even and odd functions using (1) the definition and (2) graphing calculator:
Average Rate of Change of a Function:
If a is in the domain of a function f(x), the
average rate of change of f, between a and x is defined
as
Find the rate of change of the following functions:
Answer: -4
One-to-One Functions:
A functions f is one-to-one if f(a) = f(b) only when a = b.
Horizontal line test: If no horizontal line meets the graph of a function in more than one point, the function is one-to-one.
Examples:
- f(x) = 3x +5 is one-to-one
Graph f(x) = 3x + 5 and use horizontal line test.
.
If f(a) = f(b) can you conclude that a = b?
Graph each f(x) and use horizontal line test.
Is f(x) one-to-one?
Composite Functions:
Given functions f and g the composite function fog defined by
(fog)(x)=f(g(x)).
Example: Let f(x) = 3x – 2 and g(x) = -2x + 4.
Then (fog)(x) = f(g(x))
= f(-2x + 4)
= 3(-2x + 4) – 2
= -6x + 12 – 2
= -6 x+ 10
Verify that (gof)(x) = g(f(x)) =-6x + 8
Definition of an inverse function: We say that f and g are inverse functions of each other if
f(g(x)) = x and g(f(x)) = x for all values of x in the suitable domains.
Example:
Let f(x) = 3x - 4 and g(x) = (x + 4)/3.
f(g(x)) = 3(x+4)/3 - 4 = x + 4 - 4 = x.
Similarly g(f(x)) = x.
To find the inverse of a one-to-one function f(x):
Step 1: Let y = f(x).
Step 2: Interchange x and y.
x = f(y)
Step 3: Solve x = f(y) for y. Then the solution y is the inverse function of f.
Find the inverse function of f(x) = 3x - 4.
REVIEW OF TRIGONOMETRY
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Trigonometric Functions:
For a right triangle ABC with right angle at C:
Definition: We say that an angle
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is
Trigonometric Identities:
Even-Odd Identities:
Question: Is 
an
even or odd function?
Properties of the Trigonometric Functions:
Domain
Range Period
Exercise: Fine the exact values of the following
:
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Draw a triangle with opposite side = 1 and hypotenuse = 4.Then adjacent side is square root of 15. Answer:
Use a right triangle with adj. = x, hyp. = a. Then Opp. =
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Double-Angle Formulas:
Half-Angle Formulas:
Sum and Difference Formula:
Product-to-Sum Formulas:
Sum-to-Product Formulas:
Exercises: Using the sum-to-product formula,
we get
Similarly
