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 Df and Dg of the two functions

f:  { Df x 9 x2 -sin(x)+42 x2 -2


g:  { Dg y ln(y) y2 +1 .

Exercise 6.7.2
Specify the range Wi of the function

i:  { x x2 -4x+4+π.

Exercise 6.7.3
Find the parameters A,λ in the exponential function

c:  { xA·eλx -1,

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=fg: (note: h(x)=(fg)(x)=f(g(x))) of the functions

f:  { xC·sin(x)


g:  { xB·x+π.

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 of

u:  { (0;) y- log2 (y).

The function f= u-1 has
  1. the domain Df = .
  2. the range Wf = .
  3. 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:  { [0;3) x2x+1

  ... can be also written for short as f(x)=2x+1.
  ... is a linear affine function.
  ... has the range .
  ... 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 .

Exercise 6.7.7
Calculate the following logarithms:
  1. ln(e5 · 1 e ) = .
  2. log10 (0.01) = .
  3. log2 (2·4·16·256·1024) = .

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