y= Whole question under square roots. x/x-5?

Question

find: domain,range,asymtotes,concavity,local extrema,and try to graph it

Answer 1

y = sqrt[x/(x-5)]

Start by getting a few points:
f(0) = 0 (it passes through the origin)
f(5) is undetermined (divide by zero)

See what happens when 0 < x < 5. For x = 3, x/(x-5) is negative, therefore the function is undefined over that range.

For x > 5, x/(x-5) is positive, so the function exists. When x gets very large, x >> 5 (e.g., x = 1 million), x/(x-5) approaches one, so f(x) approaches sqrt(1) = 1 from above. Therefore, y = 1 is an asymptote, but f(x) is always larger than one for very large x.

Now look at x < 0. x/(x-5) is always positive, and it's always less than one. A good point to use is:

f(-4) = sqrt[-4/(-4-5)] = sqrt(4/9) = 2/3

and another good point is

f(9) = sqrt[9/(9-5)] = sqrt(9/4) = 3/2

We know enough to plot the graph:

1. Draw the axes; draw a dotted-line asymptote at y=1; and draw another dotted-line asymptote at x=5.

2. Let's do negative x first. Start at the origin, draw the curve through (-4, 2/3), and continue the line out to the left, using y=1 as an asymptote.

3. The function is undefined for 0 < x <= 5.

4. For x>5, start the curve way up on the x=5 asymptote, bring it down through (9, 3/2), and continue it out to the right, using y=1 as an asymptote.

Answer 2

oh you funny..... domain: [-infinity,5) (5,+infinity] range: (5, +infinity) asymtotes: y = 5 no extrema... concave up

Answer 3

y = √(x / (x-5))
Domain = R - ]0 , 5]
range = [0,∞[