Question

Consider the function g(x)=x/x-7.

a) Summarize the increasing and decreasing intervals.

b) Compare the slopes of the tangents at the points where i) x=7.5 and x=20
ii) x=6.5 and x=-20

Do it for me please...

Answer 1

I have a hunch you actually mean:

g(x)=x/ (x-7)

because that is much more exciting than what you typed.

Now clearly if x = 7 we have a serious problem with this function. It has a zero for its denominator and is therefore undefined at x = 7.

Now what if x = 8 ?
g(8) = 8/1 = 8

And what if x = 6?
g(6) = 6/-1 = -6

Sketch a graph now and watch what is happening

try x = 20
g(20) = 20/13

try x = -20
g(-20) = -20/-27

try x = 100
g(100) = 100/93 humm closer and closer to 1

try x = -100
g(-100) = -100/-107 humm, closer and closer to -1

Now do the x = 7.5 and 6.5 like they said

Now slopes.
You did not say if you know calculus. If not brute force or just use your graph. If you do know then the slope is the derivative at that point so
g'(x) = [(x-7) -x ] / (x-7)^2
or
slope = -7/(x-7)^2

Answer 2

try x = -100
g(-100) = -100/-107 humm, closer and closer to +1

by the way, notice that the slope is negative everywhere.