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Section 1.3 More on Functions and Their Graphs Math 112 Section 1.3 A function is said to be increasing on an interval if, for all a and b in the interval, a > b implies f(a) > f(b). Or simply put, increasing means up, decreasing means down, and constant means level. Increasing: (4 , 1) (4 , ) Decreasing: ( , 4) Constant: (1 , 4) Relative Minimum: 2 The intervals describing where functions increase, decrease, or are constant, use x-coordinates and not the y-coordinates. Find where the graph is increasing? Where is it decreasing? Where is it constant? Example y x Example y Find where the graph is increasing? Where is it decreasing? Where is it constant? x Relative Maxima And Relative Minima Where are the relative minimums? Where are the relative maximums? Example Why are the maximums and minimums called relative or local? y x Even and Odd Functions and Symmetry Math 112 A graph is symmetric with respect to the y-axis if for any point (x , y) on the graph, the point (x , y) is also on the graph. (x , y) (x , y) Math 112 Section 1.7 A graph is symmetric with respect to the origin if for any point (x , y) on the graph, the point (x , y) is also on the graph. (x , y) (x , y) Math 112 Section 1.7 If the graph of a function f is symmetric with respect to the y-axis, (that is, f(x) = f(x)), then f(x) is an even function. If the graph of a function f is symmetric with respect to the origin, (that is, f(x) = f(x)), then f(x) is an odd function. Math 112 Even Function (x , y) Odd Function (x , y) (x , y) (x , y) Example Is this an even or odd function? y x Example Is this an even or odd function? y x Example Is this an even or odd function? y x Math 112 Even Function f(x) = 3x2 10 f(x) = 3(x)2 10 = 3x2 10 = f(x) f(x) = f(x) Odd Function f(x) = 6x3 4x f(x) = 6(x)3 4(x) = 6x3 + 4x = (6x3 4x) = f(x) f(x) = f(x) Piecewise Functions Math 112 A piecewise defined function is a function defined by different formulas for different parts of the domain. for x 4 3 f(x) 2x 1 for 4 x 4 x 2 for x 4 f(4) = 3 f(2) = 2(2) + 1 = 5 f(4) = 42 = 16 Math 112 for x 4 3 f(x) 2x 1 for 4 x 4 x 2 for x 4 21 21 A function that is defined by two or more equations over a specified domain is called a piecewise function. Many cellular phone plans can be represented with piecewise functions. See the piecewise function below: A cellular phone company offers the following plan: $20 per month buys 60 minutes Additional time costs $0.40 per minute. C t 20 if 0 t 60 20 0.40(t 60) if t>60 Example C t 20 if 0 t 60 20 0.40(t 60) if t>60 Find and interpret each of the following. C 45 C 60 C 90 Example Graph the following piecewise function. 3 if - x 3 2 x 3 if x>3 f x y x Functions and Difference Quotients See next slide. f(x+h)-f(x) 2 Find for f(x)=x 2 x 5 h First find f(x+h) f(x+h)=(x+h) 2(x+h)-5 2 x 2hx h 2 x 2h 5 2 2 Continued on the next slide. f(x+h)-f(x) Find for f(x)=x 2 2 x 5 h Use f(x+h) from the previous slide f(x+h)-f(x) Second find h 2 2 2 x 2 hx h 2 x 2 h 5 x 2 x 5 f(x+h)-f(x) h h x 2 2hx h 2 2 x 2h 5 x 2 2 x 5 h 2hx h 2 2h h h 2x h 2 h 2x+h-2 Example Find and simplify the expressions if f ( x) 2 x 1 Find f(x+h) f(x+h)-f(x) Find , h0 h Example 2 f ( x ) x 4 Find and simplify the expressions if f(x+h)-f(x) Find f(x+h) Find , h0 h Example 2 f ( x ) x 2x 1 Find and simplify the expressions if Find f(x+h) f(x+h)-f(x) Find , h0 h y There is a relative minimum at x=? Find the difference quotient for f(x)=3x 2 . Evaluate the following piecewise function at f(-1) f(x)= 2x+1 if x<-1 -2 if -1 x 1 x-3 if x>1
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