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This article provides a review of important definitions in pre-calculus, including real-valued functions, one-to-one and onto functions, even and odd functions, composition of functions, and transcendental functions.
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Math180 Review of Pre-Calculus
Definitions • A real-valued function f of a real variable x from X to Y is a correspondence (rule) that assigns to each number x X exactly one number y in Y. • The domain of f is the set X. The number y is the image of x under f and is denoted by f(x). The range of f is a subset of Y and consists of all images of numbers in X.
Definitions • A function from X to Y is one-to-one if to each y-value in the range there corresponds exactly one x-value in the domain. • A function from X to Y is onto if its range consists of all of Y.
Definitions • The function y = f(x) is even iff • The function y = f(x) is odd iff • The graph of an even function is symmetric wrt y-axis. • The graph of an odd function is symmetric wrt origin.
Definitions • Let f and g be functions. The function given by The domain of is the set of all x in the domain of g such that g(x) is in the domain of f.
Example Let Evaluate and simplify
Example • Let • Evaluate and simplify
Example • Let • Find the domain and the range.
+ - + -2 2 Domain:
x x Example • An open box of maximum volume is to be made from a square piece of material 24 inches on a side by cutting squares from the corners and turning up the sides. Express the volume V as a function of x, the length of the corner square. What is the domain of the function?
Example • Express the area of a circle as a function of its circumference.
Example • Dayton River and Light, Inc. has a power plant on the Miami River where the river is 800 feet wide. To lay a new cable from the plant to a location in the city 2 miles downstream on the opposite side cost $180 per foot across the river and $100 per foot along the land. Suppose that the cable goes from the plant to a point Q on the opposite side that is x feet from the point P directly opposite the plant. Write a function C(x) that gives the cost of laying the cable in terms of the distance x.
Example • Solve
Transcendental Functions • Exponential: where b is the base b > 0 x Reals f (x) > 0
Transcendental Functions • Logarithmic: where b is the base b > 0 x is the argument x > 0 f(x) Reals
Example • Write the expression in algebraic form:
y 1 4x
