Define $F : \mathbf { R } \times \mathbf { R } \rightarrow \mathbf { R } \times \mathbf { R }$

question 5

Essay

Define $F : \mathbf { R } \times \mathbf { R } \rightarrow \mathbf { R } \times \mathbf { R }$ as follows: $F ( x , y ) = ( 3 y - 1,1 - x )$ for all $( x , y )$ in $\mathbf { R } \times \mathbf { R }$ .
(a) $F ( 0,0 ) = ? F ( 1,4 ) =$ ?
(b) Is $F$ one-to-one? Prove or give a counterexample.
(c) Is $F$ onto? Prove or give a counterexample.
(d) Is $F$ a one-to-one correspondence? If not, explain why not.
If yes, find $F ^ { - 1 }$ .

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Definitions:

Lens

A curved piece of glass or other transparent material that focuses or refracts light, used in optical devices like glasses, cameras, and microscopes.

Retina

A thin layer of tissue at the back of the eye that contains the cells responsible for converting light into neural signals for vision.

Saturation

In color theory, the intensity of a color expressed as the degree to which it differs from white.

Amplitude

The height or strength of a wave signal, which in various contexts can describe the intensity of things like sound, light, or electrical signals.