# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 4/en

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Warm-up-exercises

### Exercise

Establish for each ${\displaystyle {}n\in \mathbb {N} }$ if the function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto x^{n},}$
is injective

and/or surjective.

### Exercise

Show that there exists a bijection between ${\displaystyle {}\mathbb {N} }$ and ${\displaystyle {}\mathbb {Z} }$.

### Exercise

Give examples of functions

${\displaystyle \varphi ,\psi \colon \mathbb {N} \longrightarrow \mathbb {N} }$
such that ${\displaystyle {}\varphi }$ is

injective, but not surjective, and ${\displaystyle {}\psi }$ is surjective, but not injective.

### Exercise

Let ${\displaystyle {}L}$ and ${\displaystyle {}M}$ be two sets and let

${\displaystyle F\colon L\longrightarrow M}$

be a function. Let

${\displaystyle G\colon M\longrightarrow L}$
be another function such that

${\displaystyle {}F\circ G=\operatorname {id} _{M}\,}$ and ${\displaystyle {}G\circ F=\operatorname {id} _{L}\,}$. Show that ${\displaystyle {}G}$ is the inverse of ${\displaystyle {}F}$.

### Exercise

Determine the composite functions

${\displaystyle \varphi \circ \psi \,\,{\text{ and }}\,\,\psi \circ \varphi }$

for the functions ${\displaystyle {}\varphi ,\psi \colon \mathbb {R} \rightarrow \mathbb {R} }$, defined by

${\displaystyle \varphi (x)=x^{4}+3x^{2}-2x+5\,\,{\text{ and }}\,\,\psi (x)=2x^{3}-x^{2}+6x-1.}$

### Exercise

Let ${\displaystyle {}L,M,N}$ and ${\displaystyle {}P}$ be sets and let

${\displaystyle F\colon L\longrightarrow M,\,x\longmapsto F(x),}$
${\displaystyle G\colon M\longrightarrow N,\,y\longmapsto G(y),}$
and
${\displaystyle H\colon N\longrightarrow P,\,z\longmapsto H(z),}$
be

functions. Show that

${\displaystyle H\circ (G\circ F)=(H\circ G)\circ F.}$

### Exercise

Let ${\displaystyle {}L,M,N}$ be sets and let

${\displaystyle f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N}$

be functions with their composition

${\displaystyle g\circ f\colon L\longrightarrow N,\,x\longmapsto g(f(x)).}$

Show that if ${\displaystyle {}g\circ f}$ is injective, then also ${\displaystyle {}f}$ is injective.

### Exercise

Let

${\displaystyle f_{1},\ldots ,f_{n}\colon \mathbb {R} \longrightarrow \mathbb {R} }$
be functions, which are increasing or decreasing, and let ${\displaystyle {}f=f_{n}\circ \cdots \circ f_{1}}$ be their

composition. Let ${\displaystyle {}k}$ be the number of the decreasing functions among the ${\displaystyle {}f_{i}}$'s. Show that if ${\displaystyle {}k}$ is even then ${\displaystyle {}f}$ is increasing and if ${\displaystyle {}k}$ is odd then ${\displaystyle {}f}$ is decreasing.

### Exercise

Calculate in the polynomial ring ${\displaystyle {}\mathbb {C} [X]}$ the product

${\displaystyle ((4+i)X^{2}-3X+9i)\cdot ((-3+7i)X^{2}+(2+2i)X-1+6i).}$

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Prove the following properties concerning the degree of a polynomial:

1. ${\displaystyle {}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}}$,
2. ${\displaystyle {}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)}$.

### Exercise

Show that in a polynomial ring over a field ${\displaystyle {}K}$ the following statement holds: if ${\displaystyle {}P,Q\in K[X]}$ are not zero, then also ${\displaystyle {}PQ\neq 0}$.

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Let ${\displaystyle {}a\in K}$. Prove that the evaluating function

${\displaystyle \psi \colon K[X]\longrightarrow K,\,P\longmapsto P(a),}$

satisfies the following properties (here let ${\displaystyle P,Q\in K[X]}$).

1. ${\displaystyle {}(P+Q)(a)=P(a)+Q(a)}$.
2. ${\displaystyle {}(P\cdot Q)(a)=P(a)\cdot Q(a)}$.
3. ${\displaystyle {}1(a)=1}$.

### Exercise

Evaluate the polynomial

${\displaystyle 2X^{3}-5X^{2}-4X+7}$
replacing the variable ${\displaystyle {}X}$ by the

complex number ${\displaystyle {}2-5i}$.

### Exercise

Perform, in the polynomial ring ${\displaystyle {}\mathbb {Q} [X]}$ the division with remainder ${\displaystyle {}{\frac {P}{T}}}$, where ${\displaystyle {}P=3X^{4}+7X^{2}-2X+5}$, and ${\displaystyle {}T=2X^{2}+3X-1}$.

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}K[X]}$ be the polynomial ring over ${\displaystyle {}K}$. Show that every polynomial ${\displaystyle {}P\in K[X],\,P\neq 0,}$ can be decomposed as a product

${\displaystyle P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q}$

where ${\displaystyle {}\mu _{j}\geq 1}$ and ${\displaystyle {}Q}$ is a polynomial with no roots (no zeroes). Moreover the different numbers ${\displaystyle {}\lambda _{1},\ldots ,\lambda _{k}}$ and the exponents ${\displaystyle {}\mu _{1},\ldots ,\mu _{k}}$ are uniquely determined apart from the order.

### Exercise

Let ${\displaystyle {}F\in \mathbb {C} [X]}$ be a non-constant polynomial. Prove that ${\displaystyle {}F}$ can be decomposed as a product of linear factors.

### Exercise

Determine the smallest real number for which the Bernoulli inequality with exponent ${\displaystyle {}n=3}$ holds.

### Exercise

Sketch the graph of the following rational functions

${\displaystyle f=g/h\colon U\longrightarrow \mathbb {R} ,}$
where each time ${\displaystyle {}U}$ is the

complement set of the set of the zeros of the denominator polynomial ${\displaystyle {}h}$.

1. ${\displaystyle {}1/x}$,
2. ${\displaystyle {}1/x^{2}}$,
3. ${\displaystyle {}1/(x^{2}+1)}$,
4. ${\displaystyle {}x/(x^{2}+1)}$,
5. ${\displaystyle {}x^{2}/(x^{2}+1)}$,
6. ${\displaystyle {}x^{3}/(x^{2}+1)}$,
7. ${\displaystyle {}(x-2)(x+2)(x+4)/(x-1)x(x+1)}$.

### Exercise

Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a polynomial with real coefficients and let ${\displaystyle {}z\in \mathbb {C} }$ be a root of ${\displaystyle {}P}$. Show that also the complex conjugate ${\displaystyle {}{\overline {z}}}$ is a root of ${\displaystyle {}P}$.

Hand-in-exercises

### Exercise (3 points)

Consider the set ${\displaystyle {}M=\{1,2,3,4,5,6,7,8\}}$ and the function

${\displaystyle \varphi \colon M\longrightarrow M,\,x\longmapsto \varphi (x),}$
defined by the following table
 ${\displaystyle {}x}$ ${\displaystyle {}1}$ ${\displaystyle {}2}$ ${\displaystyle {}3}$ ${\displaystyle {}4}$ ${\displaystyle {}5}$ ${\displaystyle {}6}$ ${\displaystyle {}7}$ ${\displaystyle {}8}$ ${\displaystyle {}\varphi (x)}$ ${\displaystyle {}2}$ ${\displaystyle {}5}$ ${\displaystyle {}6}$ ${\displaystyle {}1}$ ${\displaystyle {}4}$ ${\displaystyle {}3}$ ${\displaystyle {}7}$ ${\displaystyle {}7}$

Compute ${\displaystyle {}\varphi ^{1003}}$, that is the ${\displaystyle {}1003}$-rd composition (or iteration) of ${\displaystyle {}\varphi }$ with itself.

### Exercise (2 points)

Prove that a strictly increasing function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} }$

is injective.

### Exercise (3 points)

Let ${\displaystyle {}L,M,N}$ be sets and let

${\displaystyle f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N}$

be functions with their composite

${\displaystyle g\circ f\colon L\longrightarrow N,\,x\longmapsto g(f(x)).}$

Show that if ${\displaystyle {}g\circ f}$ is surjective, then also ${\displaystyle {}g}$ is surjective.

### Exercise (3 points)

Calculate in the polynomial ring ${\displaystyle {}\mathbb {C} [X]}$ the product

${\displaystyle ((4+i)X^{3}-iX^{2}+2X+3+2i)\cdot ((2-i)X^{3}+(3-5i)X^{2}+(2+i)X+1+5i).}$

### Exercise (3 points)

Perform, in the polynomial ring ${\displaystyle {}\mathbb {C} [X]}$ the division with remainder ${\displaystyle {}{\frac {P}{T}}}$, where ${\displaystyle {}P=(5+i)X^{4}+iX^{2}+(3-2i)X-1}$ and ${\displaystyle {}T=X^{2}+iX+3-i}$.

### Exercise (5 points)

Let ${\displaystyle {}P\in \mathbb {R} [X]}$ be a non-constant polynomial with real coefficients. Prove that ${\displaystyle {}P}$ can be written as a product of real polynomials of degrees ${\displaystyle {}1}$ or ${\displaystyle {}2}$.