Why does the reciprocal of a polynomial share the same points as the polynomial function at y = 1 and y = -1
because 1/1 = 1 and 1/-1 = -1
To understand why the reciprocal of a polynomial shares the same points as the polynomial function at y = 1 and y = -1, we need to consider the definition and properties of reciprocal functions.
The reciprocal of a number x is 1/x, which means that it is the value obtained by dividing 1 by x. Similarly, the reciprocal of a function f(x) is 1/f(x), where f(x) is the original function.
Now, let's consider a polynomial function f(x) and its reciprocal function g(x) = 1/f(x). We can analyze the behavior of the reciprocal function at y = 1 and y = -1.
1. At y = 1:
If the polynomial function f(x) attains the value of 1 at certain x-values, then its reciprocal 1/f(x) will also attain the value of 1 at those same x-values. This is because the reciprocal of 1 is still 1. Therefore, the reciprocal function will share the same points with the polynomial function at y = 1.
2. At y = -1:
Similarly, if the polynomial function f(x) attains the value of -1 at certain x-values, then its reciprocal 1/f(x) will attain the value of -1 at those same x-values. This is because the reciprocal of -1 is still -1. Therefore, the reciprocal function will share the same points with the polynomial function at y = -1.
To summarize, the reciprocal of a polynomial function shares the same points as the polynomial function at y = 1 and y = -1 because when a polynomial function f(x) attains those values at certain x-values, its reciprocal function 1/f(x) will also attain those same values at those x-values.