SOLUTION: For the function g(x)=-1+log of (x-5) to the base 2. (a) Find inverse of g(x). (b) Sketch the graph of g(x) and inverse of g(x) on the same coordinate showing the asymptotes and

Algebra ->  Graphs -> SOLUTION: For the function g(x)=-1+log of (x-5) to the base 2. (a) Find inverse of g(x). (b) Sketch the graph of g(x) and inverse of g(x) on the same coordinate showing the asymptotes and       Log On


   

Question 1084918: For the function g(x)=-1+log of (x-5) to the base 2.
(a) Find inverse of g(x).
(b) Sketch the graph of g(x) and inverse of g(x) on the same coordinate showing the asymptotes and intercepts.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

g%28x%29%22%22=%22%22-1%2Blog%282%2C%28x-5%29%29

Replace g(x) by y

y%22%22=%22%22-1%2Blog%282%2C%28x-5%29%29

Interchange x and y

x%22%22=%22%22-1%2Blog%282%2C%28y-5%29%29

Solve for y

Add 1 to both sides

x%2B1%22%22=%22%22log%282%2C%28y-5%29%29

Use the definition of a logarithm:

y-5%22%22=%22%222%5E%28x%2B1%29%29

Add 5 to both sides:

y%22%22=%22%225%2B2%5E%28x%2B1%29%29

Replace y by g-1(x)

g%5E%28-1%29%28x%29%22%22=%22%225%2B2%5E%28x%2B1%29



The red graph is of g(x), the blue graph is of g-1(x).
The green lines are the asymptotes.  The asymptote for  
g(x) has equation x=5. The asymptote for g-1(x) has 
equation y=5.  The faint blue dotted line is the graph 
of the identity function y=x.  The graph of the inverse 
of any function is the reflection of its graph across 
this identity line y=x.

Edwin