Question

What can you say about the end behavior of the function f(x)=-4x^6+6x^2-52

Answer 1

Answer:

As x gets smaller, pointing to negative infinity, the value of f decreses, pointing to negative infinity.

As x gets increases, pointing to positve infinity, the value of f decreses, pointing to negative infinity.

Step-by-step explanation:

To find the end behaviour of a function f(x), we calculate these following limits:

$\lim_{x \to +\infty} f(x)$

And

$\lim_{x \to -\infty} f(x)$

In this question:

$f(x) = -4x^{6} + 6x^{2} - 52$

At negative infinity:

$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} -4x^{6} + 6x^{2} - 52$

When the variable points to infinity, we only consider the term with the highest exponent. So

$\lim_{x \to -\infty} -4x^{6} + 6x^{2} - 52 = \lim_{x \to -\infty} -4x^{6} = -4*(-\infty)^{6} = -(\infty) = -\infty$

So as x gets smaller, pointing to negative infinity, the value of f decreses, pointing to negative infinity.

Positive infinity:

$\lim_{x \to \infty} f(x) = \lim_{x \to \infty} -4x^{6} + 6x^{2} - 52 = \lim_{x \to \infty} -4x^{6} = -4*(\infty)^{6} = -(\infty) = -\infty$

So as x gets increases, pointing to positve infinity, the value of f decreses, pointing to negative infinity.