Question

The function f ,defined by f(x) = (x-1)/(3-x) , has the same set as domain and as range. State whether the statement is true or false, justify the answer with reason.

Answer 1

Answer on Question #80552 - Math - Abstract Algebra

Question. The function $f$, defined by

$f(x)=\frac{x-1}{3-x},$

has the same set as domain and as range. State whether the statement is true or false, justify the answer with reason.

Answer. False. The range of $f$ does not contain $-1$ for the following reason. Assume that $f(x)=-1$ for some $x$. Then, as $f(x)$ has a value, $3-x\neq 0$. Hence $\frac{x-1}{3-x}=-1$, $x-1=(-1)(3-x)=x-3$, $-1=-3$, contradiction. Therefore, $f(x)\neq-1$ for all $x$.

The domain of $f$ contains $-1$ because if $x=-1$, then $3-x=4\neq 0$, so the division in the formula of $f$ is defined, and $f$ is defined at $-1$.

If the range and the domain of $f$ were the same set, then the range of $f$ would contain $-1$ because the domain of $f$ contains it. But the range of $f$ does not contain $-1$, contradiction. Therefore, the range and the domain of $f$ are not the same set.

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