Follow the ninestep graphing strategy to sketch the graph of

Follow the nine-step graphing strategy to sketch the graph of the rational function. F(x) = 2x^2 - 5x - 3/x + 1 Find the domain, Select the correct choice below and if necessary, fill in the answer box to complete your choice. The domain of f is The domain is all real numbers Does f have any removable discontinuities? There is a removable discontinuity at There are no removable disconutinties Check for symmetry. Select all that apply The graph of f is symmetric about the origin The graph of f is symmetric about the y-axis the graph has no symmetry Find the y-intercept The y-intercept is y = The function has no y-intercept Find any x-intercept The x-intercept(s) is(are) x = The function has no x-intercept Find the equations of any vertical asymptotes The vertical asymptote(s) is (are) The function has no vertical asymptotes

Solution

f(x)= (2x²-5x-3)/(x+1)

Its a rational expression and a rational expression is undefined when the denominator is 0

x+1=0

x=-1

At x=-1 the function is undefined. Therefore the domain is all the real values except x

x<-1 or x>-1

2. The given expression does not have any removable discontinuties

4. y intercept

to find the y intercept we have to plug x=0

(2(0)²-5(0)-3)/((0)+1)= -3

Therefore the y intercept is (0,-3)

5.To find the x intercept we have to plug y=0

0=(2x²-5x-3)/(x+1)

2x^2-5x-3=0

(2x+1)(x-3)=0

x=-1/2 , 3

X intercepts=(-1/2,0),(3,0)

6. Vertical asymptote

TO find the vertical asymptote we have to set denominator to zero

x+1=0

x=-1 and that\'s the vertical asymptote