S4 Flashcards
(8 cards)
Let f, g : ℝ → ℝ be two functions.
If f is injective, then f is monotonic.
False
Take f(x) = x for x ≥ 0, f(x) = 1/x for x < 0.
Let f, g : ℝ → ℝ be two functions.
If f is bijective and increasing, then its inverse f−1 is decreasing.
False:
Take f(x) = x.
Let f, g : ℝ → ℝ be two functions.
If f ◦ g is decreasing, then f and g are decreasing.
False:
Take f = x, g = −x.
The image of the circle |z| = 1 under the map f(z) = 1/z is a circle.
True:
The circle |z| = 1 is sent to itself by f.
z2 +1 divides z6 +3z4 +z2 −1.
True:
You can check that i, −i are both roots of the RHS.
Let zk, k=1,···,n be the n roots of zn +bn−1zn−1 +···+b0, then ∏zi = (−1)nb0.

True:
You can check by writing the polynomial as show.

For all complex numbers z, |z| > 1/|z| .
False:
The inequality is false everytime |z| < 1
There are infinitely many complex numbers z such that |z| = 3.
True:
The locus described by |z| = 3 is a circle.