Section 1 1 Introduction to functions Definition of a function A function f is a rule that assigns to each value x in a set D a unique value f x in a set E The set of x values is called the domain D of f The set of values of f x for x in D is called the range of f Examples of ways to define a function are 1 using a table Let D 1 2 3 4 5 x 1 2 3 4 5 f x 2 4 6 8 10 or just listing the set of ordered pairs with x in the first coordinate and f x in the 2nd 1 2 2 4 3 6 4 8 5 10 2 with a rule The above function is f x 2x for each x in D 1 2 3 4 5 We know f 3 6 by reading the table or by substituting x 3 into the formula 3 by describing in words Example A right circular cylindrical can with top and bottom must have a volume of 60 cubic cm S is the surface area of the can Find a rule for S as a function of the radius r Call the height of the can h Find S in terms of r and h Then eliminate h using the given volume The lateral surface area of the cylinder is 2 rh and the areas of the top and bottom are both r 2 So S 2 r h 2 r 2 The volume of the can is r 2 h 60 so h Substituting into S we have S r 2 r 60 r2 60 120 2 r 2 2 r 2 r r 2 The domain of the rule for S r is all real numbers except 0 But the restricted domain that fits the context is all positive real numbers Finding domains For now you need these two rules 1 We cannot take even roots of negative numbers 2 We cannot divide by 0 If a function is defined by a rule give the domain as all x values where the rule is defined Ex Find the domain of f x 16 x 2 Since we cannot take even roots of negative numbers we must have 16 x 2 0 This is true only for the interval 4 4 The square brackets mean that the endpoints are included Find the domain of each function 1 f x 2 x 2 g x 1 2 x 3 h x 1 x2 9 Answers 1 2 2 2 3 3 3 Piecewise functions In piecewise functions we follow different rules on different intervals x2 9 x 0 2 x 4 x 0 Ex f x Find f 2 f 0 f 1 2 0 so we use the first rule f 2 2 2 9 4 9 5 At 0 we also use the 1st rule since the 1st rule has 0 included f 0 9 1 0 so at 1 we use the 2nd rule f 1 6 In graphing this function we use a solid dot at the point 0 9 to indicate that f 0 9 We use a small open circle at the point 0 4 where the 2nd rule starts The open circle indicates that 0 4 is not included in the function graph Ex The absolute value function f x x is the piecewise function x f x x x 0 x 0 For example f 3 3 and f 4 4 Symmetry Even and Odd functions If f x f x for each x then f is called an even function Any constant function and any even power of x is an even function Any sum of even functions is also even Any polynomial with only even powers of x is even For example p x 5 2 x 2 x 4 is even If a function is even then its graph is symmetric about the y axis If f x f x for each x then f is called an odd function Any polynomial with only odd powers of x is an odd function For example q x x 3 x 3 4 x 5 is odd If a function is odd then its graph is symmetric about the origin The function r x 2 x x 2 is neither even nor odd Increasing and Decreasing functions If the values of f x increase as x increases f is an increasing function The graph rises as we move to the right Any line with a positive slope is an increasing function If the values of f x decrease as x increases f is a decreasing function The graph falls as we move to the right Any line with a negative slope is a decreasing function Ex f x x 2 is decreasing on the interval 0 and is increasing on the interval 0