#### Chapter 6 Elementary Functions

Section 6.7 Final Test

# 6.7.1 Final Test Module 6

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##### Exercise 6.7.1
Specify the maximum domains ${D}_{f}$ and ${D}_{g}$ of the two functions

$f:\mathrm{ }\left\{\begin{array}{ccc}\hfill {D}_{f}& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & \frac{9{x}^{2}-\mathrm{sin}\left(x\right)+42}{{x}^{2}-2}\hfill \end{array}$

and

$g:\mathrm{ }\left\{\begin{array}{ccc}\hfill {D}_{g}& \hfill \to \hfill & ℝ\hfill \\ \hfill y& \hfill ⟼\hfill & \frac{\mathrm{ln}\left(y\right)}{{y}^{2}+1} .\hfill \end{array}$

##### Exercise 6.7.2
Specify the range ${W}_{i}$ of the function

$i:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & {x}^{2}-4x+4+\pi .\hfill \end{array}$

##### Exercise 6.7.3
Find the parameters $A,\lambda \in ℝ$ in the exponential function

$c:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & A·e{}^{\lambda x}-1 ,\hfill \end{array}$

such that $c\left(0\right)=1$ and $c\left(4\right)=0$.
Answer: $A$$=$
, $\lambda$$=$
.
Simple logarithms can be left as they are, e.g. $\mathrm{ln}\left(100\right)$ can be entered as ln(100) even though the exact value of $\mathrm{ln}\left(100\right)$ is unknown.

##### Exercise 6.7.4
Specify the composition $h=f\circ g:ℝ\to ℝ$ (note: $h\left(x\right)=\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$) of the functions

$f:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & C·\mathrm{sin}\left(x\right)\hfill \end{array}$

and

$g:\mathrm{ }\left\{\begin{array}{ccc}\hfill ℝ& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & B·x+\pi .\hfill \end{array}$

Answer: $h\left(x\right)$$=$
.
Find the parameters such that the sine wave described by the function $h$ has the graph shown below.
Abbildung 1: A sine wave.

Answer: $h\left(x\right)$$=$
.

##### Exercise 6.7.5
Specify the inverse function $f={u}^{-1}$ of

$u:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left(0;\infty \right)& \hfill \to \hfill & ℝ\hfill \\ \hfill y& \hfill ⟼\hfill & -{\mathrm{log}}_{2}\left(y\right) .\hfill \end{array}$

The function $f={u}^{-1}$ has
1. the domain ${D}_{f}$$=$ .
2. the range ${W}_{f}$$=$ .
3. the mapping rule $f\left(y\right)={u}^{-1}\left(y\right)$$=$ .
Enter the ranges as intervals of the form (a;b), infinity can also be an endpoint.

##### Exercise 6.7.6

Please indicate whether the following statements are right or wrong:

The function

$f:\mathrm{ }\left\{\begin{array}{ccc}\hfill \left[0;3\right)& \hfill \to \hfill & ℝ\hfill \\ \hfill x& \hfill ⟼\hfill & 2x+1\hfill \end{array}$

 ... can be also written for short as $f\left(x\right)=2x+1$. ... is a linear affine function. ... has the range $ℝ$. ... has the slope $2$. ... can only take values greater or equal $1$ and less than $7$. ... has a graph that is a piece of a line. ... has at $x=0$ the value $1$. ... has the domain $ℝ$.

##### Exercise 6.7.7
Calculate the following logarithms:
1. $\mathrm{ln}\left(e{}^{5}·\frac{1}{\sqrt{e}}\right)$$=$ .
2. ${\mathrm{log}}_{10}\left(0.01\right)$$=$ .
3. ${\mathrm{log}}_{2}\left(\sqrt{2·4·16·256·1024}\right)$$=$ .

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