Question

Which graph shows a polynomial function,f, such that f(x)

Answer 1

A graph that shows a polynomial function,f, such that f(x) → ∞ as x → -∞ and f(x) → ∞ as x → ∞ is: D. graph D.

In Mathematics and Euclidean Geometry, the degree of a polynomial function is sometimes referred to as an absolute degree and it represents the leading coefficient of each of its term.

Based on the information provided, we can logically deduce that the leading coefficient of the above polynomial function must be positive, and have an even degree. Therefore, the end behavior can be described as follows;

• As x tends towards negative infinity, f(x) tends towards positive infinity i.e x → -∞, f(x) → ∞.
• As x tends towards positive infinity, f(x) tends towards positive infinity i.e x → ∞, f(x) → ∞.

In this context, the graph must point upward in the same direction as correctly depicted by graph D only.

Answer 2

To determine which graph shows a polynomial function f, such that

• 
  `f(x)\rightarrow \infty,` when
  `x\rightarrow -\infty;`
• 
  `f(x)\rightarrow \infty,` when
  `x\rightarrow \infty,`

you have to find the graph that is

• decreasing when x are infinitely small;
• increasing, when x are infinitely large.

Consider all options:

A. When x are infinitely small, the graph is decreasing and when x are infinitely large, the graph is decreasing too. Thus, this option is false.

B. When x are infinitely small, the graph is increasing. So, this option is false.

C. When x are infinitely small, the graph is increasing. So, this option is false.

D. When x are infinitely small, the graph is decreasing and when x are infinitely large, the graph is increasing. Thus, this option is true.

Answer: correct choice is D.