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Leya [2.2K]
2 years ago
6

title="f(x) = \frac{x^{2} +4x-4}{x^{2} -2x-8}" alt="f(x) = \frac{x^{2} +4x-4}{x^{2} -2x-8}" align="absmiddle" class="latex-formula">
Domain:
V.A:
Roots:
Y-int:
H.A:
Holes:
O.A:

Also, draw on the graph attached.

Mathematics
1 answer:
givi [52]2 years ago
7 0

i) The given function is

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

We can rewrite in factored form to obtain;

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

f(x)=\frac{(x+2\sqrt{2}+2)(x-2\sqrt{2}+2)}{(x-4)(x+2)}

The domain is

(x-4)(x+2)\ne0

(x-4)\ne0,(x+2)\ne0

x\ne4,x\ne-2

ii) To find the vertical asymptotes equate the denominator to zero.

(x-4)(x+2)=0

(x-4)=0,(x+2)=0

x=4,x=-2

iii) To find the roots, equate the numerator to zero.

(x+2\sqrt{2}+2)(x-2\sqrt{2}+2)=0}

(x+2\sqrt{2}+2)=0,(x-2\sqrt{2}+2)=0}

(x=-2\sqrt{2}-2,x=2\sqrt{2}-2)}

iv) To find the y-intercept, substitute x=0 into the equation.

f(0)=\frac{0^2+4(0)-4}{0^2-2(0)-8}

We simplify to obtain;

f(0)=\frac{-4}{-8}

f(0)=\frac{1}{2}

v) The horizontal asymptote is

lim_{\to \infty}\frac{x^2+4x-4}{x^2-2x-8}=1

The equation of the horizontal asymptote is y=1

vi) The function does not have a variable factor that is common to both the numerator and the denominator.

The function has no  holes in it.

vii) The given function is a proper rational function.

Proper rational functions do not have oblique asymptotes.

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