SOLUTION: Let f(x) = 3x-2 and g(x) = 5/(3x), for x does not equal 0. Find f^-1(x). Show that ( g ° f^-1) (x) = 5/(x+2). Let h(x) = 5/x+2, for x is greater or equal to 0. The graph of h h

Algebra ->  Functions -> SOLUTION: Let f(x) = 3x-2 and g(x) = 5/(3x), for x does not equal 0. Find f^-1(x). Show that ( g ° f^-1) (x) = 5/(x+2). Let h(x) = 5/x+2, for x is greater or equal to 0. The graph of h h      Log On


   

Question 1094396: Let f(x) = 3x-2 and g(x) = 5/(3x), for x does not equal 0.
Find f^-1(x).
Show that ( g ° f^-1) (x) = 5/(x+2).
Let h(x) = 5/x+2, for x is greater or equal to 0. The graph of h has a horizontal asymptote at y=0.
Find the y-intercept of the graph of h.
Hence, sketch the graph of h.
Find the graph of h^-1, write down the x-intercept.
For the graph of h^-1, write down the equation of the vertical asymptote.
Given that h^-1(a)=3, find the value of a.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

Let f(x) = 3x-2 and g(x) = 5%2F%283x%29, for x does not equal 0.
Find f-1(x).

1. In place of f(x), replace it by y

y = 3x-2

2. Interchange x and y

x = 3y-2

3. Solve for y
   x = 3y-2
3y-2 = x          <--swap sides
  3y = x+2
   y = %28x%2B2%29%2F3

4. In place of y, replace it by f-1(x)

f-1(x) = %28x%2B2%29%2F3  


Show that (g∘f-1)(x) = 5%2F%28x%2B2%29.

g(x) = 5%2F%283x%29

To avoid so many fractions, rewrite using "÷"

g(x) = matrix%281%2C3%2C5%2C%22%F7%22%2C3x%29

In place of x, replace it by the right side of f-1(x),
which is {{(x+2)/2}}}

(g∘f-1)(x) = matrix%281%2C3%2C5%2C%22%F7%22%2C3%28%28x%2B2%29%2F3%29%29%29

(g∘f-1)(x) = matrix%281%2C3%2C5%2C%22%F7%22%2Ccross%283%29%28%28x%2B2%29%2Fcross%283%29%29%29%29
(g∘f-1)(x) = matrix%281%2C3%2C5%2C%22%F7%22%2Cx%2B2%29

(g∘f-1)(x) = 5%2F%28x%2B2%29

Let h(x) = 5%2F%28x%2B2%29, for x≧0. The graph of h has a horizontal
asymptote at y=0.
Find the y-intercept of the graph of h. 
We let x equal to 0

h(0) = 5%2F%280%2B2%29
h(0) = 5%2F2

So the y-intercept is the point %28matrix%281%2C3%2C0%2C%22%2C%22%2C5%2F2%29%29

Hence, sketch the graph of h. 

Since x≧0, the graph starts at the y-intercept and goes to the
right.  The graph does not extend left of the y-intercept:

 

You do the rest by yourself.  If you have trouble, tell me in
the thank you note form below, and I will get back to you by
email. 

Edwin