1.3
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Section 1.5 More on Functions and Their Graphs Increasing and Decreasing Functions The open 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 Example Find where the graph is increasing? Where is it decreasing? Where is it constant? y 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 A graph is 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. All even functions have graphs with this kind of symmetry. A graph is symmetric with respect to the origin if, for every point (x,y) on the graph, the point (-x,-y) is also on the graph. Observe that the first- and thirdquadrant portions of odd functions are reflections of one another with respect to the origin. Notice that f(x) and f(-x) have opposite signs, so that f(-x)=-f(x). All odd functions have graphs with origin symmetry. 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 Piecewise Functions 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 f x if - x 3 y 2 x 3 if x>3 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 Find and simplify the expressions if Find f(x+h) f ( x) x 4 2 f(x+h)-f(x) 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 Some piecewise functions are called step functions because their graphs form discontinuous steps. One such function is called the greatest integer function, symbolized by int(x) or [x], where int(x)= the greatest integer that is less than or equal to x. For example, int(1)=1, int(1.3)=1, int(1.5)=1, int(1.9)=1 int(2)=2, int(2.3)=2, int(2.5)=2, int(2.9)=2 Example The USPS charges $ .42 for letters 1 oz. or less. For letters 2 oz. or less they charge $ .59, and 3 oz. or less, they charge $ . 76. Graph this function and then find the following charges. a. The charge for a letter that weights 1.5 oz. y b. The charge for a letter that weights 2.3 oz. $1.00 $ .75 $ .50 $ .25 x y There is a relative minimum at x=? (a) 4 (b) 3 (c) 2 (d) 0 Find the difference quotient for f(x)=3x 2 . (a) 6 (b) 3 x 2 6 xh (c) 6 x h (d) 6 x Evaluate the following piecewise function at f(-1) 2x+1 if x<-1 f(x)= (a) 2 (b) 4 (c) 0 (d) 1 -2 if -1 x 1 x-3 if x>1
