Book of Proof - Amazon S3

212 Functions
We can check our work by confirming that g −1 (g(m, n)) = (m, n). Doing
the math,
g −1 (g(m, n)) = g −1 (m + n, m + 2n)
= ( 2(m + n) − (m + 2n),(m + 2n) − (m + n) )
= (m, n).
Exercises for Section 12.5
1. Check that the function f : Z → Z defined by f (n) = 6 − n is bijective. Then
compute f −1 .
2. In Exercise 9 of Section 12.2 you proved that f : R − { 2 } → R − { 5 } defined by
f (x) = 5x + 1 is bijective. Now find its inverse.
x − 2
3. Let B = { 2 n : n ∈ Z } = { ..., 1 4 , 1 2 ,1,2,4,8,...} . Show that the function f : Z → B
defined as f (n) = 2 n is bijective. Then find f −1 .
4. The function f : R → (0,∞) defined as f (x) = e x3 +1 is bijective. Find its inverse.
5. The function f : R → R defined as f (x) = πx − e is bijective. Find its inverse.
6. The function f : Z × Z → Z × Z defined by the formula f (m, n) = (5m + 4n,4m + 3n)
is bijective. Find its inverse.
7. Show that the function f : R 2 → R 2 defined by the formula f (x, y) = ((x 2 + 1)y, x 3 )
is bijective. Then find its inverse.
8. Is the function θ : P(Z) → P(Z) defined as θ(X) = X bijective? If so, what is its
inverse?
9. Consider the function f : R × N → N × R defined as f (x, y) = (y,3xy). Check that
this is bijective; find its inverse.
10. Consider f : N → Z defined as f (n) = (−1)n (2n − 1) + 1
. This function is bijective
4
by Exercise 18 in Section 12.2. Find its inverse.
12.6 Image and Preimage
It is time to take up a matter of notation that you will encounter in future
mathematics classes. Suppose we have a function f : A → B. If X ⊆ A, the
expression f (X) has a special meaning. It stands for the set { f (x) : x ∈ X } .
Similarly, if Y ⊆ B then f −1 (Y ) has a meaning even if f is not invertible.
The expression f −1 (Y ) stands for the set { x ∈ A : f (x) ∈ Y } . Here are the
precise definitions.