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Find the domain and the range of the rea...

Find the domain and the range of the real function, `f(x)=(x-3)/(x-5)`.

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We have, `f(x)=(x-3)/(x-5)`.
Clearly, f(x) is defined for all real values of x except that at which `x-5=0,i.e.,x=5`.
`:."dom"(f)=R-{5}`
Let `y=f(x)`. Then,
`y=(x-3)/(x-5)impliesxy-5y=x-3`
`impliesx(y-1)=5y-3impliesx=(5y-3)/(y-1)." "....(i)`
It is clear from (i) that x is not defined when y-1=0, i.e., when y=1.
`:."range"(f)=R-{1}`.
Hence, dom `(f)=R-{5}" and range "(f)=R-{1}`.
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