S4

Question 1

Let f, g : ℝ → ℝ be two functions.

If f is injective, then f is monotonic.

Answer 1

False

Take f(x) = x for x ≥ 0, f(x) = ¹/_x for x < 0.

Question 2

Let f, g : ℝ → ℝ be two functions.

If f is bijective and increasing, then its inverse f⁻¹ is decreasing.

Answer 2

False:

Take f(x) = x.

Question 3

Let f, g : ℝ → ℝ be two functions.

If f ◦ g is decreasing, then f and g are decreasing.

Answer 3

False:

Take f = x, g = −x.

Question 4

The image of the circle |z| = 1 under the map f(z) = ¹/_z is a circle.

Answer 4

True:

The circle |z| = 1 is sent to itself by f.

Question 5

z² +1 divides z⁶ +3z⁴ +z² −1.

Answer 5

True:

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

Question 6

Let z_{k, k=1,···,n} be the n roots of zⁿ +b_n−1zⁿ⁻¹ +···+b₀, then ∏z_i = (−1)ⁿb₀.

Answer 6

True:

You can check by writing the polynomial as show.

Question 7

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

Answer 7

False:

The inequality is false everytime |z| < 1

Question 8

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

Answer 8

True:

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