6.7.1 Final Test Module 6

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 : {\begin{matrix} D_{f} & \to & ℝ \\ x & ⟼ & \frac{9 x^{2} - \sin (x) + 42}{x^{2} - 2} \end{matrix}

and

g : {\begin{matrix} D_{g} & \to & ℝ \\ y & ⟼ & \frac{\ln (y)}{y^{2} + 1} . \end{matrix}

Exercise 6.7.2

Specify the range

W_{i}

of the function

i : {\begin{matrix} ℝ & \to & ℝ \\ x & ⟼ & x^{2} - 4 x + 4 + π . \end{matrix}

Exercise 6.7.3

Find the parameters

A, λ \in ℝ

in the exponential function

c : {\begin{matrix} ℝ & \to & ℝ \\ x & ⟼ & A \cdot e^{λ x} - 1, \end{matrix}

such that

c (0) = 1

and

c (4) = 0

.
Answer:

A

=

λ

=

.
Simple logarithms can be left as they are, e.g.

\ln (100)

can be entered as ln(100) even though the exact value of

\ln (100)

is unknown.

Exercise 6.7.4

Specify the composition

h = f \circ g : ℝ \to ℝ

(note:

h (x) = (f \circ g) (x) = f (g (x))

) of the functions

f : {\begin{matrix} ℝ & \to & ℝ \\ x & ⟼ & C \cdot \sin (x) \end{matrix}

and

g : {\begin{matrix} ℝ & \to & ℝ \\ x & ⟼ & B \cdot x + π . \end{matrix}

Answer:

h (x)

=

.
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 (x)

=

Exercise 6.7.5

Specify the inverse function

f = u^{- 1}

u : {\begin{matrix} (0; \infty) & \to & ℝ \\ y & ⟼ & - \log_{2} (y) . \end{matrix}

The function

f = u^{- 1}

has

the domain $D_{f}$ $=$ .
the range $W_{f}$ $=$ .
the mapping rule $f (y) = u^{- 1} (y)$ $=$ .

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 : {\begin{matrix} [0; 3) & \to & ℝ \\ x & ⟼ & 2 x + 1 \end{matrix}

		... can be also written for short as $f (x) = 2 x + 1$ .
		... 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:

$\ln (e^{5} \cdot \frac{1}{\sqrt{e}})$ $=$ .
$\log_{10} (0.01)$ $=$ .
$\log_{2} (\sqrt{2 \cdot 4 \cdot 16 \cdot 256 \cdot 1024})$ $=$ .

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Onlinebrückenkurs Mathematik

1. Elementary Arithmetic

2. Equations in one Variable

3. Inequalities in one Variable

4. System of Linear Equations

5. Geometry

6. Elementary Functions

7. Differential Calculus

8. Integral Calculus

9. Objects in the Two-Dimensional Coordinate System

10. Basic Concepts of Descriptive Vector Geometry

11. Language of Descriptive Statistics