Let $S$ be the set of all nonzero real numbers. Define a function $g$ from $S$ to $S$ by the formula $g ( x ) = \frac { 1 } { x }$, for all nonzero real numbers $x$.
(a) Show that $g$ is a one-to-one correspondence from $S$ to $S$.
(b) Find $g ^ { - 1 }$.

Question

Quizplus · Accepted Answer

The Answer is a. Proof that g is one-to-one: Suppose x1 and x2 are any nonzero real numbers and
g(x1) = g(x2). [We must show that x1 = x2.]
By deﬁnition of g,

\frac { 1 } { x _ { 1 } } = \frac { 1 } { x _ { 2 } }

Cross-multiplying gives that

x _ { 1 } = x _ { 2 }

[as was to be shown]. Proof that g is onto: Let y be any nonzero real number. [We must show that there is a nonzero real number whose image under g is y.]

Let SSS Be the Set of All Nonzero Real Numbers ggg From SSS

Let $S$ Be the Set of All Nonzero Real Numbers $g$ From $S$