F(x) = x^2+x-2/x^2-3x-4 what is the domain and range, x-and y-intercepts, horizontal asymptotes, and vertical asymptotes?

i. Domain and Range

The given function is

The domain of this function is,

The range refers to the y-values for which x is defined. x  is defined for all values of y.

The range is all real numbers. See graph

ii. x-and-y-intercept

For x- intercept intercept we put

This implies that;

This will give us

The x-intercepts are

For y-intercept, we put

to obtain;

The y-intercept is

iii. Horizontal asyptote

Since degree of the numerator and the denominator are the same, there is a horizontal asymptote

To find the horizontal asymptote.

We divide the leading coefficient of the numerator by the leading  coefficient of the denominator.

The horizontal asymptote is

iv. Vertical asymptote

To find the vertical asymptote, we equate the denominator to zero to get;

This implies that;

Split the middle term

Factor

Factor further

The vertical asymptotes are

D : {x ∈ R : x≠-1 and x≠4}

R : {all real number}

The x intercepts are 2,-1    

The y intercept is 1/2

The horizontal asymptotes is y=1

x=-1  x=4

Step-by-step explanation:

f(x) = x^2+x-2/x^2-3x-4

We need to factor

f(x) = (x+2) (x-1)

     ( x-4) (x+1)

The denominator goes to zero at 4 and -1

The domain is the possible values for x.  X cannot make the denominator go to zero so it cannot be -1 or 4

D : {x ∈ R : x≠-1 and x≠4}

The range is the possible values for y.  It can take on any real numbers

R : {all real number}

The x intercepts is where it crosses the x axis.  This is where the numerator goes to zero

The x intercepts are 2,-1

The y intercepts are where it crosses the x axis.  Let x =0 and solve for y

y = (0+2) (0-1)               -2

        = = 1/2

     ( 0-4) (0+1)            -4

The y intercept is 1/2

The horizontal asymptotes is 1

Since the polynomial is the same degree in the numerator and the denominator,divide the coefficients

1/1 = 1

y=1

Vertical asymptoptes are where the denominator is zero

x=-1  x=4

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step-by-step explanation: