Find the following information, if any. If the graph does not have any, state there are none.

Question

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Find the following information, if any. If the graph does not have any, state there are none.

𝑅(𝑥) = (5𝑥 − 10)/(2𝑥 + 4)

a) What is the domain of 𝑅(𝑥)? Give your answer in set-builder notation.

b) The equation of any vertical asymptote(s).

c) The equation of any horizontal asymptote.

d) The 𝑥-intercept(s). Show some work!

e) The 𝑦-intercept. Show some work!

f) If x is -6 then what is R(x).

g) If x is 1 then what is R(x).

h) Give an accurate sketch of the graph of 𝑅(𝑥). Include the asymptotes as dashed lines and any holes on the graph.

Answer 1

𝑅(𝑥) = (5𝑥 − 10)/(2𝑥 + 4)

Find the following information, if any. If the graph does not have any, state there are none.

𝑅(𝑥) = (5𝑥 − 10)/(2𝑥 + 4)

a) What is the domain of 𝑅(𝑥)? Give your answer in set-builder notation. The denominator cannot be equal to zero. If you set this equal to zero and solve, the is the break in the domain.

2𝑥 + 4 = 0 Subtract 4 from both sides

2x=-4 Divide both sides by 2

x=-2

Domain: {x | x<-2 or x>-2}

b) The equation of any vertical asymptote(s). The vertical asymptote is where the denominator equals zero, we have already found this. In the equation form it would be x=-2

c) The equation of any horizontal asymptote. Since the exponent of the numerator equals the exponent of the denominator, the horizontal asymptote is the x axis or y=0

d) The 𝑥-intercept(s). Show some work! Set the numerator equal to zero and solve for the x intercepts.

5𝑥 − 10 = 0 Add 10 to both sides

5x=10 Divide both sides by 5

x=2

The point would be (2,0)

e) The 𝑦-intercept. Show some work! Set x=0 and solve

𝑅(0) = (5𝑥 − 10)/(2𝑥 + 4)

𝑅(0) = (5(0) − 10)/(2(0) + 4) Simplify

𝑅(0) = -10/4 Reduce

𝑅(0) = -5/2

The point would be (0,-5/2)

f) If x is -6 then what is R(x).

𝑅(-6) = (5(-6) − 10)/(2(-6) + 4) Simplify

𝑅(-6) = 5

g) If x is 1 then what is R(x).

𝑅(1) = (5(1) − 10)/(2(1) + 4) Simplify

𝑅(1) = -5/6

h) Give an accurate sketch of the graph of 𝑅(𝑥). Include the asymptotes as dashed lines and any holes on the graph. Take all of the information in parts a-g, plot them and create the graph.