Pre-Calculus (Part I) - Module 1 :
Functions And Their Properties
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Function (01.01) ANS✔✔ A function f from a set A to a set B is a rule of
correspondence that assigns to each element x in the set A exactly one element
y in the set B. The set A is the domain (or set of inputs) of the function f, and
the set B contains the range (or set of outputs).
Characteristics of Functions (01.01) ANS✔✔ 1. For each value in the
domain, there is only one (1) corresponding value in the range.
2. All elements in the domain must be used.
3. More than one value in the domain can be matched with the same value in
the range.
Domain of a Function (01.01) ANS✔✔ The domain of a function can be all
the valid values for the expression that defines the function. For example, the
domain of f(x) = 1 - x is all real numbers. For f(x) = 1 over quantity 1 minus x
end quantity the domain would be all real numbers except 1 (division by zero
is invalid). Also, since a domain is a set of real numbers, the domain of f(x) =
√x is x greater than or equal to 0.
Euler's Function Notation (Section 1.2) ANS✔✔ y = f(x) (read as "y equals f
of x" or "the value of f at x"
, x = the independent variable; y = the dependent variable
The Graph of a Function (Section 1.2) ANS✔✔ The set of all points (x,f(x) or
y), x in the domain of f. Domain values along the x-axis are matched with their
corresponding range values along the y-axis to give ordered pairs to make the
graph.
Vertical Line Test (Section 1.2) ANS✔✔ A graph is defined as a function only
if a vertical line intersects the graph in only one point. If it intersects in more
than one place at a time, it is not a function.
Types of Functions (Section 1.3) ANS✔✔ (First Six Are Most Common
Algebraic Functions)
1. The Constant Function: f(x) = b (output is always the same no matter the
domain, like y = 7)
2. The Identity Function: f(x) = x
3. The Absolute Value Function: f(x) = |x| = abs (x)
4. The Square Root Function: f(x) = square root of x
5. The Squaring Function: f(x) = x^2
6. The Cubing Function: f(x) = x^3 (origin is often called the "point of
inflection" as the graph change curvature at this point)
7. The Reciprocal Function: f(x) = 1/x
8. The Exponential Function: f(x) = e^x
9. The Natural Logarithm Function: f(x) = ln x
10. The Sine Function: f(x) = sin x
11. The Cosine Function: f(x) = cos x
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