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

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) = ?
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1 Answer

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)

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