Question

What is the end behavior of the function $f(x) = {x^3} + 2{x^2} + 4x + 5?$

Answer 1

Hint: The end behavior of the polynomial function can be determined by the term of the highest degree. Degree is identified by the highest power in the polynomial and then will identify the end behavior of the function. Here we will place the infinite value and then simplify it for the required expression.

Complete step-by-step answer:
For the large values for $x$ first identify the highest degree of the polynomial which is much larger than the other terms.
Here, as we can observe, ${x^3}$is the term with the highest power and it is positive and the degree is odd.
Thus, the end behavior can be stated as –
When $x \to \infty \Rightarrow f(x) \to + \infty $ and
Similarly, when $x \to - \infty \Rightarrow f(x) \to - \infty $
Hence, the end behavior for the given polynomial can be from $ - \infty $to $ + \infty $

Note: Polynomial can be defined as the expression having two or more terms. Do not get confused between the terms of the degree and the power and apply it accordingly. The power can be represented by the number which is raised to another number called the exponent whereas the degree can be represented by the order of the polynomial. Power is used to represent the long mathematical expression in the short form.