Where positive, negative or zero?
Sign and Zero Analysis of functions
See or remember these geometric and algebraic previews of calculus (Chapters 2 to 4 in Volume 3) for an easy informal introduction to
- the monotonicity of functions y = f(x), that is where they are increasing or decreasing, with the where referring to intervals,
- the location of extrema: local, end-point and absolute maxima and minima, and
- the sign analysis not of functions, but of formulas for their slopes.
That should make the following definitions easy to understand and the following exercises
For a Curve or function y
For A Quadratic (factored)
For a Cubic Factored
Un-Sign Functions etc.
easy to do or complete.
Sign of Functions
In the following, suppose y = f(x) is real-valued function of a real variable x whose domain includes a set S.
- Definition: f is non-positive on S if and only if for
each real number a in S, f(a) < 0
Example of a function f(x) which is non-positive - that is, its values are negative or zero.
- Definition: f is negative on S if and only if for
each real numbers a in S, f(a) < 0
Example of a function negative on the drawn part of its graph.
- Definition: f is non-negative on S if and only
if for each real numbers a in S, f(a) >
0
Example of a non-negative - positive or zero everywhere. - Definition: f is positive on S if and only if for each
real numbers a in S, f(a) > 0.
Example of a function positive on an interval [c,d]
In our examples, the set S may be an interval in the domain of f, an interval that might be finite, infinite, semi-finite, and in the finite or semi-finite cases may or may not include end-points.
Remark: Sign and zero analysis of a function y = f(x) determines, if possible, the maximal intervals on which the function is non-negative or non-positive, and also locates the zeroes of f(x). That is, those points or intervals where f(x) = 0. The points x where y = f(x) = 0 give the x-intercepts of the function f or its graph.
Exercises
Now try the following exercises
For a Curve or function y
For A Quadratic (factored)
For a Cubic Factored
Un-Sign Functions etc.