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1 year ago

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Can someone help me with this? I’ve been struggling with it for a couple days and it’s due today. I’ll give brainliest if you an

swer.

1 answer:

Lynna [10]1 year ago

6 0

Answer:

ammm sige and I will be back from tomorrow so I'll be in the leader of my class

Step-by-step explanation:

sige and sa will of in in in couple couple

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1 year ago

Show all work to identify the asymptotes and zero of the function f(x) = 6x / x^2 - 36

eduard

Answer:

Zero of the function f(x) is at x = 0

Vertical Asymptotes at x = ±6

Horizontal Asymptotes at y = 0

Step-by-step explanation:

<h3>Vertical Asymptotes </h3>

For a given function f(x):

Vertical Asymptotes are obtained at those values of x, where the function f(x) tends to infinity, I.e.,

<em>When</em><em> </em><em>x</em><em> </em><em>approaches</em><em> </em><em>some</em><em> </em><em>constant</em><em> </em><em>value</em><em> </em><em>b</em><em>u</em><em>t</em><em> </em><em>th</em><em>e</em><em> </em><em>curve</em><em> </em><em>moves</em><em> </em><em>towards</em><em> </em><em>infinity</em><em>.</em><em> </em>

If f(x) is a fraction, it'll tend to infinity when it's denominator becomes zero.

Vertical Asymptotes of the given function can be obtained by walking thru the following steps:

<u>Step I</u>

(Factorise the numerator and denominator)

$\mathsf{ f(x) = \frac{6x}{ {x}^{2} - 36 } }$

<em>x</em><em>²</em><em> </em><em>-</em><em> </em><em>36</em><em> </em><em>can</em><em> </em><em>be</em><em> </em><em>facto</em><em>rised</em><em> </em><em>into</em><em> </em><em>(</em><em>x</em><em> </em><em>+</em><em> </em><em>6</em><em>)</em><em>(</em><em>x</em><em> </em><em>-</em><em> </em><em>6</em><em>)</em>

<em>and</em><em>,</em><em> </em><em>ofcourse</em><em>,</em><em> </em><em>we</em><em> </em><em>can</em><em> </em><em>write</em><em> </em><em>6</em><em>x</em><em> </em><em>as</em><em> </em><em>6</em><em>(</em><em>x</em><em> </em><em>-</em><em> </em><em>0</em><em>)</em><em> </em>

$\mathsf{ f(x) = \frac{6(x - 0)}{ (x + 6)(x - 6) } }$

<u>Step</u><u> </u><u>II</u>

(Reduce the fraction to its simplest form by canceling out the common factors)

<em>There aren't any common factors in the numerator and denominator in this case.</em>

<u>Step</u><u> </u><u>III</u>

(Look for the values of x which cause the denominator to be zero)

<em>If</em><em> </em><em>we</em><em> </em><em>put</em><em> </em>x = 6

<em>denominator</em><em> </em><em>becomes</em><em> </em><em>0</em>

Also,

<em>If</em><em> </em><em>we</em><em> </em><em>substitute</em><em> </em><em>x</em><em> </em><em>with</em><em> </em> -6

<em>denominator</em><em> </em><em>becomes</em><em> </em><em>0</em><em>.</em><em> </em>

The two values of x indicate the two Vertical Asymptotes of the function f(x).

Therefore,

<u>Vertical</u><u> </u><u>Asymptotes</u><u> </u><u>of</u><u> </u><u>the</u><u> </u><u>given</u><u> </u><u>function</u><u> </u><u>f</u><u>(</u><u>x</u><u>)</u><u> </u><u>are</u><u>:</u>

$\boxed{ \mathsf{x = \pm6}}$

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

<h3 /><h3>Horizontal Asymptotes:</h3>

Horizontal Asymptotes are obtained When x tends to infinity and y approaches some constant value.

I'll be using the concept of limits for this.

$\mathsf{y = \frac{6x}{ {x}^{2} - 36 } }$

<em>dividing</em><em> </em><em>and</em><em> </em><em>multiplying</em><em> </em><em>by</em><em> </em><em>x</em><em>²</em><em> </em><em>(</em><em>Yep</em><em>!</em><em> </em><em>so</em><em> </em><em>if</em><em> </em><em>x</em><em> </em><em>becomes</em><em> </em><em>infinity</em><em> </em><em>1</em><em>/</em><em> </em><em>x</em><em> </em><em>and</em><em> </em><em>1</em><em>/</em><em> </em><em>x</em><em>²</em><em> </em><em>all</em><em> </em><em>such</em><em> </em><em>terms</em><em> </em><em>become</em><em> </em><em>0</em><em>,</em><em> </em><em>'</em><em>cause</em><em> </em><em>1</em><em>/</em><em> </em><em>∞</em><em> </em><em>is</em><em> </em><em>0</em><em>)</em><em> </em>

$\implies \mathsf{y = lim_{x \rightarrow \infty }( \frac{ \frac{6x}{ {x}^{2} } }{ \frac{ {x}^{2} - 36 }{ {x}^{2} } } ) }$

$\implies \mathsf{y = lim_{x \rightarrow \infty }( \frac{ \frac{6}{ x } }{ 1- \frac{36 }{ {x}^{2} } } ) }$

Substitute x with ∞, you get zero/ 1

$\implies \boxed{\mathsf{y = 0}}$

So, the horizontal Asymptote of the function is y = 0, that is the x axis

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

<h3>Zeroes of a function:</h3>

The values of x that reduces f(x) to zero are called the zeroes of f(x).

Here, only x = 0 acts as the zero of the function.

[NOTE:

For finding <u>Vertical Asymptotes</u><u>,</u>Equate the denominator to 0. And
For finding <u>Zeroes</u><u>,</u> Equate the numerator to 0]

__________________

[That's what it's graph looks like. ]

3 0

2 years ago