Math140 Properties of Functions

1.6 PROPERTIES OF FUNCTIONS Even and Odd Functions A function that is symmetric with the y-axis is called even. In other words, a function f is even if for every number x in its domain the number –x is also in the domain and f(-x) = f(x) A function that is symmetric with the origin is called odd. In other words, a function f is odd if for every number x in its domain the number –x is also in the domain and f(-x) = -f(x). Note a function symmetric with the x-axis is not odd. In fact, a function symmetric with the x-axis is not a function of x at all, because it does not pass the vertical line test. Always verify your conjectures about functions with the definitions that f(-x) = f(x) for even functions and f(x)= -f(x) for odd functions. Example 2 Use a graphing utility to conjecture whether the following functions are even, odd, or neither. State symmetry. a) f(x) = x2-5 Check if even: f(-x) = f(x) ? (-x)2 – 5 = x2 -5 yes Since it is an even function, it is symmetric with respect to the y-axis. 1.5 Obtaining Information from the Graph of a Function 4 5, 3 3 -4, 2 2 -2, 1 1 0 -5 -4 -3 -2 -1 0, 0 0 4, 0 1 2 3 4 6, 0 5 6 7 -1 -2 2, -2 -3 a) b) c) d) e) f) g) h) i) j) k) l) m) n) Find f(0) and f(6) What is y when x is 0 and x is 6? From the data given, we see the y-coordinate at x=0 is 0, so f(0)=0. The y-coordinate at x=6 is also 0, so f(6)=0. Find f(2) and f(-2) What is y when x is 2 and x is -2? From the data given, we see the y-coordinate at x=2 is -2, so f(2)=-2. The y-coordinate at x=-2 is 1, so f(-2)=1. Is f(3) positive or negative? We see that at x=3 the graph is below the x-axis (where y < 0) so f(3) is negative. Is f(-1) positive or negative? We see that at x=3 the graph is below the x-axis (where y < 0) so f(3) is negative. For what numbers x is f(x) = 0? In other words, at which values of x cross the x-axis (where y=0)? The graph crosses the x-axis at x=0,x=4, x=6. For what numbers x is f(x) <0? In other words, at which values of x is the graph below the x-axis? Remember, the coordinates where y=0 are not included. The graph is < 0 only for 0 < x < 4. In interval form this is (0,4). What is the domain of f? Domain is the possible x values. Remember that this graph does not continue into infinity on both sides. It is only defined for the graph drawn. Therefore, we can infer that the possible x values are -4 ≤ x ≤ 6, or [-4,6] What is the range of f? The y values range from as low as -2 to as high as 3, so range is {y| -2 ≤ y ≤ 3} or [-2,3]. What are the x-intercepts? The x-intercepts are found when y=0, which are the points {(0,0), (4,0),(6,0)}. What is the y-intercept? By definition, this would not be a function if it crossed the y-axis (or any other vertical line) more than once. The only point that does this is (0,0). How often does the line y=-1 intersect the graph? If we draw a horizontal line through y=-1, we’d see it intersects twice. How often does he line x=1 intersect the graph? Three times. For what value of x does f(x) = 3? Remember f(x) is the same as y. What is x when y=5? There’s only one point on the graph that gives a y-value of 3. That is when x=5. For what value of x does f(x)=-2? There’s only one point on the graph that gives a y-value of -2. That is when x=2. Example 3 x f ( x) = x+2 a) Is the point (1, ½) on the graph of f? Substitute 1 for x and ½ for f(x) and see if the statement is true. Does ½ = 1/(1+2)? ½ ≠ ⅓ Therefore (1, ½) is not on the graph. b) If x = -1, what is f(x)? f(-1) = -1/(-1+2) = -1/1 = -1 The point at x=-1 is (-1,-1). c) If f(x) = 2, what is x? YOU DO THIS! Example 4 – Average Cost Function 20,000 C ( x) = 0.56 x − 34.39 x + 1212.57 + x 2 Where C(x) = average cost and x = number of computers per day. a) Determine average cost for 30 computers. C (30) = 0.56(30) 2 − 34.39(30) + 1212.57 + 20,000 = $1351.54 30 d) Graph the function C= C(x), 0 f(2) -3 Local Minimum f(2)= -2 A function f is increasing on a open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1)< f(x2). A function f is decreasing on a open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1)> f(x2). A function f is constant on a open interval I if, for any choice of x1 and x2 in I, we have f(x1)= f(x2). (The values f(x) are equal for all choices of x in I.) A function f has a local minimum of f at x=c if there is an open interval I containing c so that for all x ≠ c in I, f(x) > f(c). We call f(c) a local maximum of f. A function f has a local maximum of f at x=c if there is an open interval I containing c so that for all x ≠ c in I, f(x) < f(c). We call f(c) a local maximum of f. These are called “local” because there may be other points outside of the interval I that are less than the local minimum or greater than the local maximum. 7 Example 4 f(x) 6 5 4 3 2 1 0 -4 -3 -2 -1 0 1 2 3 4 5 6 -1 a) b) c) d) e) At what number(s) does f have a local maximum? There is an interval -11 has a local mimum at x=2. What are the local minima? The local minima ar f(-1) = 1 and f(3)=0. List the intervals on which f is increasing and on which f is decreasing. f is increasing at -13 Æ(-1,1) and (3, ∞ ) f is decreasing at x<-1 and at 10 and always decreasing if m<0. Domain is all real numbers, Range is all real numbers. Constant Function: f(x) = b where b is a real number. This is a horizontal line going through the y-intercept=b. Domain is all real numbers. Range is the single number, b. Identity Function: f(x) = x This is a line with slope = 1 and y-intercept = 0. Function is always increasing. Domain is all real numbers. Range is all real numbers. Square Function: f(x) = x2 This is an upward parabola with vertex at the origin. It is an even function which is decreasing for the interval (− ∞,0) and increasing on the interval (0, ∞ ) Domain is all real numbesrs. Range is {y|y>0} Cube Function: f(x) = x3 This is an odd function whose y-intercept is at (0,0). Function is always increasing. Domain is all real numbers. Range is all real numbers. Square Root Function: f ( x) = x This is a function that is increasing on the interval x> 0. Domain is {x|x>0} Range is {y|y>0} Reciprocal Function: f(x) = 1/x This is an odd function with no intercepts. The function is undefined at x=0 and is decreasing on the intervals (− ∞,0) and (0, ∞ ) Domain is {x|x≠0} Range is {y|y≠0} Absolute Value Function: f(x) = |x| This function is the line f(x) = -x for x < 0 and then changes to f(x) = x for x > 0. This is an even function that is decreasing for x<0 and increasing for x>0. Domain is all real numbers. Range is {y|y > 0}