Determine the end behavior of each rational function below.
f(x) = 7x5 − 3x + 1 / 4x3 + 2

When x ⇒ +∞ and x ⇒ -∞, f(x) ⇒ +∞

Step-by-step explanation:

Hi there!

First, let´s write the function:

f(x) = (7x⁵ - 3x +1) / (4x³ + 2)

When x ⇒ +∞

f(x) ⇒ (7x⁵ - 3x) / 4x³    because (7x⁵ - 3x)>>>1 and 4x³>>>2 we can neglect those constants.

f(x) ⇒7x⁵/4x³ - 3x/4x³

f(x) ⇒ 1.75x² - 0.75/x²    (0.75/x² ⇒ 0 because a number divided by a very big number is approximately zero). Then:

f(x) ⇒ 1.75x² ⇒ +∞

When x ⇒ -∞

f(x) ⇒ 1.75x² = +∞ (because x² is always positive)

Then, when x ⇒ +∞ and x ⇒ -∞, f(x) ⇒ +∞

The answer is yes. Shapes with parallel sides means that both lines are of equal distance within each other. This means that if you extend the lines up to an infinite value, the two lines never intersect. Shapes with congruent sides, however, means that both lines have the same measure or value. When we connect the end points of the lines, we can picture out the shape of this quadrilateral to be a square, rectangle, rhombus, or any regular quadrilateral for that matter.