fx 4 5xx2 3x 10 Find the domain Range Xintercept Yinterc

f(x) = 4 - 5x/x^2 - 3x - 10 Find the domain. Range. X-intercept Y-intercept. Vertical asymptote. Horizontal asymptote. Is the horizontal or slant asymptote intercept. Explain. As x approaches infinity, what does y approaches? As x approaches any vertical asymptote, what does y approaches?

Solution

f(x) = ( 4 - 5x ) / ( x^2 - 3x - 10 )

factoring the denominator

( 4 - 5x ) / ( x -5 )(x+2)

so domain is all values of x except x = 5 , -2

interval notation ( - infinity , -2) U ( -2 , 5 ) U ( 5 , infinity )

b) range

range is all real values of y that is (-infinity , + infinity )

c) for x intercept , plug f(x) = 0 and solve for x

0 = ( 4 - 5x ) / ( x^2 - 3x - 10 )

4 - 5x = 0

x = 4/5

x intercept is (4/5,0)

d) for y intercept plug x = 0 in the function

f(x) = ( 4 - 5(0)) / ( (0)^2 - 3(0) - 10 ) = 4 / -10 = - 2/ 5

y intercept is ( 0 , -2/5)

e) set denominator to zero and solve for x forvertical asymptote

( x+2) ( x- 5) = 0

x = -2 , x = 5   ( vertical asymptotes )

f) horizontal asymptote

since degree of numerator is less than degree of denominator the horizontal asymptote is y = 0