Find a formula for f ^-1(x), the inverse of the function shown below. Identify the domain and range of f ^-1

Question

Find a formula for f ^-1(x), the inverse of the function shown below. Identify the domain and range of f ^-1

 

F(x) =      3     

          x -  4                       

 

F^-1(x) = ?

Answer 1

Hi Keith;

f(x)=3/(x-4)

The denominator cannot equal zero.  The domain is from negative infinity to positive infinity except for that which corresponds to...

x-4=0

x=4

(-infinity, 4)U(4, +infinity)

Because the results of this equation can never be zero, the range is...

(-infinity, 0)U(0, +infinity)

f^-1(x)=

y=3/(x-4)

Let's isolate x...

Let's cross-multiply...

y(x-4)=3

Let's divide both sides by y...

[y(x-4)]/y=3/y

x-4=3/y

Let's add 4 to both sides...

x-4+4=(3/y)+4

x=(3/y)+4

Let's switch x and y...

y=(3/x)+4

f^-1(x)=(3/x)+4

Domain is...

(-infinity, 0)U(0, +infinity)

Because the denominator x cannot equal zero, f^-1(x) cannot result in 4.

Range is...

(-infinity, 4)U(4, +infinity)