University:
College
Course:
College Mathematics
Academic year:

2024
Views:

295

Pages:

5
Author:

Giselle A.

1.5 PROPERTIES OF FUNCTIONS When is a function increasing, decreasing, or constant? f(x) 4 Local Maximum f(5)=3 3 f(x) is constant for -4 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 2 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}