Functions Definition: A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write x y. A relation can be expressed by an equation, graph, mapping (a map) or ordered pairs. Definition: For nonempty sets X and Y, a function, f, from X to Y is a relation that associates with each element of X exactly one element of Y. The set X is called the domain of the function. For each x X, the corresponding element y Y is called the value of the function at x or the image of x. Denoted f(x). The set of all possible values of f(x) is called the range of the function f. Note: Since there may be some elements in Y that are not the image of some x X, it follows that the range of a function may be a subset of Y. Example: Determine if the relation represents a function. If so, state the domain & range. R = {(1, 4), (, 5), (3, 6), (4, 7)}. ***This relation is a function because there are no ordered pairs with the same first element and different second element. The domain is the set {1,, 3, 4} and the range is the set {4, 5, 6, 7}. Example: Determine if the relation represents a function. If so, state the domain & range. R = {( 3, 9), (, 4), (0, 0), ( 3, 8)}. ***This relation is not a function because there are ordered pairs with the same first element and different second element: ( 3, 9) and ( 3, 8). Example: f(x) = x is a function from the real numbers to real numbers. The domain is (, ) since x is defined for all real numbers. The range is [0, ) since x can never be negative. ***Notation: Sometimes, instead of writing f(x) = x, we define a function by writing y = x. Example: y = ± x is not a function since a function can only assign one value to each x. Example: Write each domain in interval notation. y = 1 x Domain: (, ). y = 1 1 x Domain: (, 1) (1, ) since undefined at x = 1. y = 1 x Domain: (, 1]. f(x) = Domain: (1, ). 1
Definition: To evaluate a function, f(x), at a given value simply replace each x with the given value. Example: Evaluate each function at the given value. For f(x) = 3x 3 1, evaluate at x =. f() = 3() 3 1 = 3 8 1 = 4 1 = 3 For f(x) = x 3x, evaluate at x = 1. f( 1) = ( 1) 3( 1) = 4 = Definition: If f and g are functions, then the sum f+g is the function defined by (f + g)(x) = f(x) + g(x) The domain of f + g consists of the numbers x that are in the domains of both f and g, i.e. dom(f + g)=dom(f) dom(g) Definition: The f-g is the function defined by (f g)(x) = f(x) g(x) with dom(f g)=dom(f) dom(g) Definition: The f g is the function defined by (f g)(x) = f(x) g(x) with dom(f g)=dom(f) dom(g) Definition: The f g is the function defined by ( f f(x) g )(x) = g(x), where g 0 with dom( f g )=dom(f) dom(g) {x dom(g) g(x) 0}. Example: Let f and g be two functions defined by f(x) = x (f + g)(x) (f g)(x) (f g)(x) ) (x) ( f g and g(x) = x + 3. Find (f + g)(x) = x + x + 3 (f g)(x) = x x 3 (f g)(x) = x (x + 3) = x3 +3x ( f g )(x) = x x+3 = x 1 x+3 = x x +x 3
Graphing Functions Definition: The graph of a function f is the set of all points (x, y) such that y = f(x). The height y is the function s value f(x). Theorem: (Vertical-Line Test) A curve is the graph of a function if and only if no vertical line intersects it more than once. Example: Which curve is the graph of a function? Only Graph 1 is a function Example: Find the domain and range of the function. What is the value at f(0)? Domain: [-3,-3] Range: [0,] f(0) = A graph is said to be symmetric with respect to the x-axis if, for every point (x, y) on the graph, the point (x, y) is also on the graph. A graph is said to be symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point ( x, y) is also on the graph. A graph is said to be symmetric with respect to the origin if, for every point (x, y) on the graph, the point ( x, y) is also on the graph. 3
Definition: f increases on an open interval I if the value of f(x) increases as x moves from left to right on I. f decreases on I if f(x) decreases as x goes from left to right on I. f is constant on I if the values of f(x) are equal for all choices of x in I. f has a local maximum at c if there is an open interval I containing c so that for all x in I, f(x) f(c). We call f(c) a local maximum value of f. f has a local minimum at c if there is an open interval I containing c so that for all x in I, f(x) f(c). We call f(c) a local minimum value of f. If there is a number c in the domain I of f for which f(x) f(c) for all x in I, then f(c) is the absolute maximum of f. If there is a number c in the domain I of f for which f(x) f(c) for all x in I, then f(c) is the absolute minimum of f. Theorem (Extreme Value Theorem) If f is a continuous* function whose domain is a closed interval [a, b], then f has an absolute maximum and an absolute minimum on [a, b]. *We won t give a precise definition of a continuous function, but we will agree for now that a continuous function is one whose graph has no gaps or holes and can be traced without lifting the pencil from the paper. We ve already discussed the slope of a line as the rate of change of the line, and now we ll extend that notion to functions in general. Definition: If a and b with a b are in the domain of the function y = f(x), then the average rate of change of f from a to b is: Average rate of change = y x = f(b) f(a) b a **Note: In the above definition, the symbol represents change in. So y is the change in y, and x is the change in x. Example: Find the average rate of change of f(x) = x from 1 to 3. y x = f(3) f(1) 3 1 = (3) (1) = 8 = 4 Example: Find the average rate of change of f(x) = x from 1 to 5. y x = f(5) f(1) 5 1 = (5) (1) 4 = 4 4 = 6 Theorem: (Slope of the Secant Line) The average rate of change of a function, f, from a to b equals the slope of the secant line containing the two points (a, f(a)) and (b, f(b)). 4
Definition: Sometimes a function is defined differently on different parts of its domain. When functions are defined by more than one equation, they are called piecewise-defined functions. Examples: (graphed in class) f(x) = { x if x 0, x if x > 0, x if x < 0, g(x) = 1 if x = 0, x if 0 < x 9. Exercise: For the function, x + 1 if 3 x < 1, h(x) = if x = 1, x if x > 1, 1. determine the domain and range of h, and. graph h. 5