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Transcript
Math 35 2.5 "An Introduction to Functions" Objectives: * De…ne function, domain, and range. * Identify functions. * Use the vertical line test. * Use function notation. * Find the domain of a function. * Graph linear functions. De…ne Function, Domain, and Range De…nition: Function A function is a set of ordered pairs in which no two ordered pairs have the same …rst coordinates and di¤erent second coordinates. > The domain of a relation is the set of all …rst coordinates of the ordered pairs. > The range of a relation is the set of all second coordinates of the ordered pairs. Example 1: (Finding the domain and range) Find the domain and range of the relation f(5; 6) ; ( 12; 4) ; (8; 6) ; ( 6; 6) ; (5; 4)g: Since we will often work with sets of ordered pairs of the form (x; y), it is helpful to de…ne a function using the variables x and y: Identify Functions Determining if " y is a Function of x" Given a relation in x and y, if to each value of x in the domain there corresponds exactly one value of y in the range, then y is said to be a function of x. In the previous de…nition, since y depends on x, we call x the independent variable and the y the dependent variable. Example 2: (Determining if y is a function of x) In each case, …nd the domain and range and determine whether the relation de…nes y to be a function of x. a) b) Page: 1 x y 3 5 0 2 3 1 Notes by Bibiana Lopez Intermediate Algebra by Tussy and Gustafson 2.5 Example 3: (Determining if y is a function of x) Determine whether each equation de…nes y to be a function of x: b) x2 + y 2 = 36 a) x + y = 5 Use the Vertical Line Test The Vertical Line Test kIf a vertical line intersects a graph in more than one point, the graph is not the graph of a function.k Example 4: (Using the vertical line test ) Determine whether each of the following is the graph of a function. y 2 a) y b) -2 2 -2 1 -4 -2 x 2 4 x -1 Use Function Notation Function Notation: The notation For example: WARNING! indicates that y is a function of x: f (x) = 2x 5 The symbol f (x) denotes a function. Example 5: (Finding the output of a function) x2 + 2x Let f (x) = : Find: 2 a) f (2) It does not mean f x (f times x) b) f (w + 2) Page: 2 Notes by Bibiana Lopez Intermediate Algebra by Tussy and Gustafson 2.5 Example 6: (Finding the output of a function) Let g (x) = 2x2 5x 4: Find: b) g t2 a) g ( 4) 1 Example 7: (Finding the input when the output is giving) x Let f (x) = + 4: For what value of x is f (x) = 5? 3 Find the Domain of a Function We can think of a function as a machine that takes some input x and turns it into some output f (x). The set of numbers that we put into the machine is the domain of the function, and the set of numbers that comes out is the range. Example 8: Find the domain of each function. a) f (x) = 2x 6 b) g (t) = 2 t2 + 3t + 2 Page: 3 c) h (s) = js 3j Notes by Bibiana Lopez Intermediate Algebra by Tussy and Gustafson 2.5 Graph Linear Functions A linear function is a function that can be written in the form f (x) = mx + b : Its graph is a straight line with slope m and y intercept (0; b) : De…nition: "Identity Function" The most basic linear function is called the identity function because it assigns each real number to itself. De…nition: "Constant Function" A linear function de…ned by is called a constant function, because for any input x, the output is the constant b (horizontal line). Example 9: (Graphing a line) Graph: f (x) = 3x 2 y 4 x f (x) = 3x 2 2 -4 -2 2 -2 4 x -4 Page: 4 Notes by Bibiana Lopez