What are the zeros of a function and how to define them?
What are zeros of a function? The answer is pretty simple - it's a mathematical term, by which is meant the domain of the definition of a given function, on which its value is zero. The zeros of a function are also called the roots of the equation. The easiest way to clarify what zeros of a function are, on a few simple examples.
Examples
Consider the simple equation y = x + 3. Since the zero of a function is the value of the argument at which y has acquired a zero value, substitute 0 to the left side of the equation:
0 = x + 3;
x = -3.
In this case, -3 is the desired zero. For this function, there is only one root of the equation, but this is not always the case.
Let's consider another example:
y = x2-9.
We substitute 0 to the left side of the equation, as in the previous example:
0 = x2-9;
-9 = x2 .
It is obvious that in this case the zeros of the function will betwo: x = 3 and x = -3. If there was an argument of the third degree in the equation, there would be three zeros. One can draw a simple conclusion that the number of roots of a polynomial corresponds to the maximum degree of the equation in the equation. However, many functions, for example y = x3 , at first glance contradict this statement. Logic and common sense suggest that this function has only one zero at the point x = 0. But in fact there are three roots, they all just coincide. If the equation is solved in a complex form, this becomes obvious. x = 0 in this case, the root whose multiplicity is 3. In the previous example, the zeros did not coincide, therefore they had multiplicity 1.
The algorithm for determining
From the examples presented, you can see how to determine the zeros of a function. The algorithm is always the same:
- Write a function.
- Substitute y or f (x) = 0.
- Solve the resulting equation.
The complexity of the last item depends on the degreeargument of the equation. When solving high-degree equations, it is especially important to remember that the number of roots of the equation is equal to the maximum power of the argument. This is especially true for trigonometric equations, where dividing both parts by a sine or cosine leads to a loss of roots.
Equations of an arbitrary degree are most easily solved by the Gorner method, which was developed specifically for finding the zeros of an arbitrary polynomial.
The meaning of the zeros of functions can be eithernegative, and positive, real or lying in the complex plane, single or multiple. Or the roots of the equation may not be. For example, the function y = 8 will not get a zero value for any x, because it does not depend on this variable.
The equation y = x2-16 has two roots, and both lie in the complex plane: x1= 4i, x2= -4і.
Common Mistakes
A common mistake that schoolchildren allow, stillnot really understood what are the zeros of a function, is the replacement of the argument (x) by zero, and not the value (y) of the function. They are surely substituted into the equation x = 0 and, based on this, y is found. But this is a wrong approach.
Another mistake, as already mentioned,sine or cosine in the trigonometric equation, because of which one or more zeros of the function are lost. This does not mean that nothing can be reduced in such equations, simply with further calculations it is necessary to take into account these "lost" factors.
Graphical representation
To understand what zeros of a function are, you can usemathematical programs, such as Maple. In it, you can draw a graph, indicating the desired number of points and the desired scale. The points at which the graph intersects the OX axis are the desired zeros. This is one of the fastest ways to find the roots of a polynomial, especially if its order is higher than the third. So if there is a need to regularly perform mathematical calculations, to find the roots of polynomials of arbitrary degrees, to build graphs, Maple or a similar program will simply be indispensable for the implementation and verification of calculations.
