Question

ac{2x-1}{x^{2}-x-6 }" alt="f(x) = \frac{2x-1}{x^{2}-x-6 }" align="absmiddle" class="latex-formula">
Domain:
V.A:
Roots:
Y-Int:
H.A:
Holes:
O.A:

Also draw on the graph.

Answer 1

i) The given function is

$f(x)=\frac{2x-1}{x^2-x-6}$

We factor to obtain

$f(x)=\frac{2x-1}{(x-3)(x+2)}$

The domain is

$(x-3)(x+2)\ne0$

$(x-3)\ne0,(x+2)\ne0$

$x\ne3,x\ne-2$

ii) The vertical asymptotes are

$(x-3)(x+2)=0$

$(x-3)=0,(x+2)=0$

$x=3,x=-2$

iii) To find the root, we equate the numerator to zero.

$2x-1=0$

$x=\frac{1}{2}$

iv) To find the y-intercept, put x=0 into the function.

$f(0)=\frac{2(0)-1}{(0)^2-(0)-6}$

$f(0)=\frac{-1}{-6}$

$f(0)=\frac{1}{6}$

vi) To find the horizontal asymptote, we take limit to infinity.

This implies that;

$lim_{x\to \infty}\frac{2x-1}{x^2-x-6}=0$

The horizontal asymptote is y=0.

vii) The numerator and the denominator do not have common factors that are at least linear.

Therefore the function has no holes in it.