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Flashcards in S4 Deck (8)
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1

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. 

2

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.

3

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. 

4

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. 

5

  z2 +1 divides z6 +3z4 +z2 −1. 

True:

You can check that i, −i are both roots of the RHS. 

6

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.

7

For all complex numbers z, |z| > 1/|z| . 

False:

The inequality is false everytime |z| < 1 

8

There are infinitely many complex numbers z such that |z| = 3. 

True:

The locus described by |z| = 3 is a circle.