S4 Flashcards

(8 cards)

1
Q

Let f, g : ℝ → ℝ be two functions.

If f is injective, then f is monotonic.

A

False

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

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2
Q

Let f, g : ℝ → ℝ be two functions.

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

A

False:

Take f(x) = x.

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3
Q

Let f, g : ℝ → ℝ be two functions.

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

A

False:

Take f = x, g = −x.

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4
Q

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

A

True:

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

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5
Q

z2 +1 divides z6 +3z4 +z2 −1.

A

True:

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

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6
Q

Let zk, k=1,···,n be the n roots of zn +bn−1zn−1 +···+b0, then ∏zi = (−1)nb0.

A

True:

You can check by writing the polynomial as show.

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7
Q

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

A

False:

The inequality is false everytime |z| < 1

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8
Q

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

A

True:

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

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