Math180 Review of Pre-Calculus 1

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. 2

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. 3

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. 4

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. 5

Example Let Evaluate and simplify

Example Let Evaluate and simplify

Example Let Find the domain and the range.

Domain:

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. x x 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.

Trig Identities

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

Transcendental Functions

Example Write the expression in algebraic form:

y 4x 1