Temperature Readings The equation to convert the temperature from degrees Fahrenheit to degrees Celsius is: c(x) = (x - 32) The equation to convert the temperature from degrees Celsius to degrees Fahrenheit is: f(x) = x + 32 Is there a temperature that has the same reading in both Fahrenheit and Celsius?

Function Set of ordered pairs {(x,y)| x  X, y  Y}, where every element of X is associated with a unique element of Y. X is the domain (set of inputs) of the function. Y is the range of the function. The image is the set of outputs.

Some Functions to Remember Equal Functions: f(x) = g(x) Identity Function: f(x) = x, id R (7) = 7 Constant Function: f(x) = 3, k(x) = y 0 Absolute Value Function: y = |x|

Describing Functions List of ordered pairs Rule Table Graph Function Diagram Verbal Description

Examples of Function Rules f(x) = -2x + 1 f(x) = x f(x) = 8 f(x) = x f(x) = 2 x Associate each integer with a number that is twice the integer.

Composite Functions The composition of g with f is the function g o f = g o f(x) = g(f(x)) Notice that g o f is obtained by first doing f and then doing g.

Properties of Some Functions One-to-one A function is one-to-one if it never sends two elements of the domain to the same element of the range. Onto A function is onto if no element of the range goes unused.

One-to-One or Onto? Temperature functions: c(x) = (x - 32) f(x) = x + 32 f(x) = |x|, Domain = R, Range = R f(x) = x 2, Domain = R, Range = R A function that assigns each word in the English to the first letter in the word. A function that assigns each real number with a point on the number line. y = 2x, Domain = Z, Range = Z

Inverses in Mathematics Inverse Property Additive Inverse, Multiplicative Inverse (reciprocal) Inverse Operation Inverse Function If a function has an inverse function, then it is 1-1. If a function is 1-1, then it has an inverse function. g -1 (g(x)) = g (g -1 (x)) = x, or g -1 o g = g o g -1 = id(x)

Find the inverse function of each of these functions: y = 2x y = -3x + 5 y = x + 32 y = x 2