Practice!

Sample Problem

Find the inverse of f(x)=(2x+3)/(x-5).

a)f-1(x)=(x+)/(x-)

 

b)What is the domain of f(x)?

All real numbers except for x=

 

c)What is the range of f(x)?

All real numbers except for y=

 

d)What is the domain of f-1(x)?

All real numbers except for x=

 

e)What is the range of f-1(x)?

All real numbers except for y=

Solution

a)f(x)=(2x+3)/(x-5)=y

(x-5)y=2x+3, xy-5y=2x+3, xy-2x=3+5y, x(y-2)=3+5y

x=(3+5y)/(y-2)

Interchange x and y:

y=(3+5x)/(x-2)=f-1(x)

b) The domain of f(x)= (2x+3)/(x-5); all real numbers except for whatever makes denominator 0, x=5

c) Range of f(x)= (2x+3)/(x-5); there’s a horizontal asymptote at y=2 (if the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients)

d) The domain of f(x)=range of f-1(x)=5 and

e) The range of f(x)=domain of f-1(x)=2