Exponential and Logarithmic Functions.

Question

Find the domain of f,range of f,inverse function, and asymtotes

domain of f^-1,range of f^-1, and draw the graphs of f and f^-1
f (x) = 1/2 log (2x)

Answer 1

Step 1:

The function is .

Power rule of logarithms: .

Rewrite the function using above formula.

Domain of logarithm function is defined for or in interval notation.

Domain of  is .

Range of the logarithm function is defined as .

Range of  is .

Vertical asymptote of the logarithmic function is .

Therefore Vertical asymptote of is or -axis.

Answer 2

Continued:

Step 2:

Find the inverse function of .

Consider .

The inverse function is defined implicitly by the equation .

Solve for .

Common logarithm function: if and only if .

If , then .

Therefore the inverse function is  .

Find the domain and range of .

Domain of is and range is .

Here , so the domain  of is and

Range of is .

Answer 3

Continued:

Step 3:

Graph:

Graph the function .

 

Graph the inverse function .

.

Solution:

Domain set of  is .

Range set of  is .

Inverse function of   is  .

Vertical asymptote of is or -axis.

Domain set of  is .

Range set of is .