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Teaching aids and simulators in the Integral online store for grade 11
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical Constructor 6.1"
Guys, we just have to consider one more method for solving equations - functional-graphical. The essence of the method is simple, and we have already used it.
Let us be given an equation of the form $f(x)=g(x)$. We construct two graphs $y=f(x)$ and $y=g(x)$ on the same coordinate plane and mark the points at which our graphs intersect. The abscissa of the intersection point (x coordinate) is the solution to our equation.
Since the method is called functional-graphical, it is not always necessary to build graphs of functions. You can also use the properties of functions. For example, you see an explicit solution to an equation at some point: if one of the functions is strictly increasing and the other is strictly decreasing, then this will be the only solution to the equation. The properties of monotonicity of functions often help in solving various equations.
Let us recall another method: if on the interval X, the largest value of any of the functions $y=f(x)$, $y=g(x)$ is equal to A, and, accordingly, the smallest value of the other function is also equal to A, then the equation $f( x)=g(x)$ is equivalent to the system: $\begin (cases) f(x)=A, \\ g(x)=A. \end (cases)$
Example.
Solve the equation: $\sqrt(x+1)=|x-1|$.
Solution.
Let's build graphs of functions on the same coordinate plane: $y=\sqrt(x)+1$ and $y=|x-1|$.
As can be seen from the figure, our graphs intersect at two points with coordinates: A(0;1) and B(4;3). The solution to the original equation will be the abscissas of these points.
Answer: $x=0$ and $x=4$.
Example.
Solve the equation: $x^7+3x-134=0$.
Solution.
Let's move on to the equivalent equation: $x^7=134-3x$.
You can see that $x=2$ is a solution to this equation. Let's prove that this is the only root.
Function $y=x^7$ – increases throughout the entire domain of definition.
Function $y=134-3x$ – decreases over the entire domain of definition.
Then the graphs of these functions either do not intersect at all, or intersect at one point, we have already found this point $x=2.$
Answer: $x=2$.
Example.
Solve the equation: $\frac(8)(x)=\sqrt(x)$.
Solution.
This equation can be solved in two ways.
1. Again, note that $x=4$ is the root of the equation. On the segment $)
