Question

According to the graph of the rational function y equals 4 over the quantity x squared minus 4 end quantity which of the following statements is/are true? The function is even. The function is increasing for all values in the domain. There is a horizontal asymptote along the x-axis. I only I and II only I and III only I, II, and III

Answer 1

Using function concepts, it is found that the correct options are:

I and III only

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The function is:

$y = \frac{4}{x^2 - 4}$

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Statement 1:

A function is even if: $f(x) = f(-x)$

We have that:

$f(x) = \frac{4}{x^2 - 4}$

$f(-x) = \frac{4}{(-x)^2 - 4} = \frac{4}{x^2 - 4} = f(x)$

Since $f(x) = f(-x)$, the function is even, and the statement is true.

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Statement 2:

The function increases when: $f^{\prime}(x) > 0$

The derivative is:

$f^{\prime}(x) = \frac{-8x}{(x^2-4)^2}$

The denominator is always positive, but the numerator can be both positive/negative, which means that when the numerator is negative(x > 0), the derivative will be negative, thus the function will decrease and the statement is false.

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Statement 3:

A horizontal asymptote is given by:

$y = \lim_{x \rightarrow \infty} f(x)$

In this question:

$y = \lim_{x \rightarrow \infty} \frac{4}{x^2 - 4} = \frac{4}{\infty - 4} = \frac{4}{\infty} = 0$

y = 0 is the x-axis, thus, the statement is true, and the correct option is:

I and III only

A similar problem is given at https://brainly.com/question/23535769

Answer 2

Answer:

I and III

Step-by-step explanation: