Answer to Question #172679 in Calculus for Todd Phillips

Question

Given y = 2x – 4 / x+1

a. State whether the function is odd or even.

b. Find the vertical and horizontal asymptotes.

c. Find the x- and y-intercepts.

d. Find y’ and y’’, intervals of increase and decrease, inflection points and concavity.

e. Sketch the function.

Answer 1

Solution

a.      Substitution x = -x into function expression gives y = f(-x) = –2x – 4 / (–x+1). Therefore f(x) ≠ f(-x) and f(x) ≠ -f(-x). So the function is not odd and not even.

b.      lim_x→±–1f(x)=±∞ => x = -1 is vertical asymptote. lim_x→±∞f(x)=±∞ => there is no horizontal asymptote.

c.      For y = 0, 2x – 4 /( x+1) = 0 => 2x²+2x–4 = 0 => x_1,2 = (–2±√(4+32))/4 = (-2±6)/4 => Two points of intersection of x-axis: x₁ = 1, x₂ = -2. For x = 0 y = –4. It is the point of intersection of y-axis. 

d.      y’ = 2 + 4 / (x+1)² . y’’ = –8/(x+1)³ . y’ > 0 for any x. => Interval of increase is (-∞,∞). y’’ ≠ 0 => No inflection points.

y’’>0 and y’ positive and increasing for x<-1 => Function is concave up for x<-1.

y’’<0 and y’ positive and decreasing for x>-1 => Function is concave down for x>-1.

e.

Answer is highlighted in bold.