Question

How do you find the end behavior of \[f(x) = - {x^2}(1 - 2x)(x + 2)\]?

Answer 1

Hint: Here the given question need us to get the value of the function or the behavior of the function at positive and negative extremities, which means when we put the value as positive infinity and negative infinity then the question need us to check the value of function for these extremities.

Complete step-by-step answer:
The given question is \[f(x) = - {x^2}(1 - 2x)(x + 2)\]
Here we have to check the value of the function for positive infinity and negative infinity:
On solving first for positive infinity, we see that the variable “x” is here three times, the first time it is in square form which means the positive infinity will remain as positive, now for the second time “x” is in “-2x” form which means positive infinity will become negative here, and finally third time is in the form of simple ”x” hence here no change in value.
Now plotting everything in mathematical form we get:
\[
   \Rightarrow for\,x = + \infty ,\,we\,get: \\
   \Rightarrow {x^2} = {\infty ^2} = \infty \\
   \Rightarrow 1 - 2x = 1 - 2(\infty ) = - \infty \\
   \Rightarrow x + 2 = \infty + 2 = \infty \\
   \Rightarrow f(\infty ) = - (\infty )( - \infty )(\infty ) = \infty \\
  hence\,for\,x \to \infty ,\,f(x) \to \infty \\
 \]
Now similarly we have to solve for the negative extremity that is for “x” as minus infinity, on solving we get:
\[
   \Rightarrow for\,x = - \infty ,\,we\,get: \\
   \Rightarrow {x^2} = - {\infty ^2} = \infty \\
   \Rightarrow 1 - 2x = 1 - 2( - \infty ) = \infty \\
   \Rightarrow x + 2 = - \infty + 2 = - \infty \\
   \Rightarrow f(\infty ) = - (\infty )(\infty )( - \infty ) = \infty \\
  hence\,for\,x \to - \infty ,\,f(x) \to \infty \\
 \]
Here values at extremities are the same for both the positive and negative extremities, and the value is plus infinity.

Note: The given question can also be solved by plotting the curve for the given function, for which we need to expand the expression and then by assuming different values of “x” we can get “f(x)”, with the help of theses coordinates we can plot the graph and see the behavior at extremities.