- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.8 Exercises
In each of Exercises 3 through 12, a function $T: V_2 \rightarrow V_2$ is defined by the formula given for $T(x, y),$ where $(x, y)$ is an arbitrary point in $V_2.$ In each case, determine whether $T$ is one-to-one on $V_2.$ If it is, describe its range $T(V_2);$ for each point $(u, v)$ in $T(V_2),$ let $(x, y) = T^{-1}(u, v)$ and give formulas for determining $x$ and $y$ in terms of $u$ and $v.$
10.$\quad$ $T(x, y) = (x + 1, y + 1).$
Solution. $\quad$ Let $A = (x_0, y_0)$ and $B = (x_1, y_1)$ be points in $V_2$ with transformations $T(A) = (x_0 + 1, y_0 + 1)$ and $T(B) = (x_1 + 1, y_1 + 1).$ Then, if $T(A) = T(B),$ we have $x_0 + 1 = x_1 + 1$ and $y_0 + 1 = y_1 + 1.$ In other words, $T(A) = T(B)$ implies $A = B,$ making $T$ a one-to-one transformation on $V_2.$ The range of $T$ is $V_2$ and for each point $(u, v)$ in $T(V_2)$ we have $x = u - 1$ and $y = v - 1. \quad \blacksquare$