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9000007604

Level:

Find the domain of the function $f(x) = 1 + \left | \frac{1} {|x|+1}\right |$.

$\mathbb{R}$

$\mathbb{R}\setminus \{ - 1\}$

$\mathbb{R}\setminus \{ - 1;1\}$

$\mathbb{R}\setminus \{ - 1;0;1\}$

$\mathbb{R}\setminus \{1\}$

Jane has to reach the opposite shore of a lake. She has three possibilities how to cross. She can use her own boat, start immediately and sail at the velocity $4\, \mathrm{km}/\mathrm{h}$. Another option is to wait for her friend, Peter, with a faster boat. Peter's boat is capable to sail at the velocity $10\, \mathrm{km}/\mathrm{h}$. However, his boat will be available in $1.5$ hours from now. The last option is to use the regular passenger transportation line which will departure in $2.25$ hours from now and sails at the speed $20\, \mathrm{km}/\mathrm{h}$. Find the interval of distances to the opposite shore for which the fastest option is to use Peter's boat.

between $10$ and $15$ kilometers

up to $10$ kilometers

between $15$ and $20$ kilometers

larger that $20$ kilometers

9000007605

Level:

Find the domain of the function $f(x) = 1 + \left | \frac{1} {-|x|+1}\right |$.

$\mathbb{R}\setminus \{ - 1;1\}$

$\mathbb{R}\setminus \{ - 1\}$

$\mathbb{R}\setminus \{ - 1;0;1\}$

$\mathbb{R}\setminus \{1\}$

$\mathbb{R}$

9000007608

Level:

Find the range of the function $f(x) = 1 + \left | \frac{1} {2(x-2)}\right |$.

$(1;\infty )$

$\mathbb{R}$

$\mathbb{R}\setminus \{1\}$

$(0;\infty )$

$\mathbb{R}\setminus \{2\}$

9000007609

Level:

Find the range of the function $f(x) = 1 + \left | \frac{1} {|x|+1}\right |$.

$(1;2] $

$\mathbb{R}\setminus \{ - 1;1\}$

$(0;\infty )$

$(1;\infty )$

$\mathbb{R}$

9000007610

Level:

Find the range of the function $f(x) = 1 + \left | \frac{1} {-|x|+1}\right |$.

$(1;\infty )$

$\mathbb{R}\setminus \{ - 1\}$

$(0;\infty )$

$\mathbb{R}\setminus \{ - 1;1\}$

$\mathbb{R}$

9000008008

Level:

Given the functions \[ \text{$f(x) = -\frac{2} {x}$ and $g(x)= \frac{k} {x}$} \] find the value of the parameter $k\in \mathbb{R}\setminus \{0\}$ which ensures \[ g(2) = 2f(-2). \]

$4$

$2$

$- 1$

$- 2$

9000007202

Level:

Consider the function \[ f(x) = [x] + 3 \] on the domain $\mathop{\mathrm{Dom}}(f) = (1;2)$. Find the parameters $a$ and $b$ and a domain of the linear function \[ g\colon y = ax + b \] which ensure that $f$ and $g$ are identical functions. \[ \] Hint: The function $y = [x]$ is a floor function: the largest integer less than or equal to $x$. For positive $x$ it is also called the integer part of $x$.

$a = 0$, $b = 4$; $\mathop{\mathrm{Dom}}(g) = (1;2)$

$a = 0$, $b = 3$; $\mathop{\mathrm{Dom}}(g) = (1;2)$

$a = 3$, $b = 0$; $\mathop{\mathrm{Dom}}(g) = (1;2)$

$a = -3$, $b = 0$; $\mathop{\mathrm{Dom}}(g) = (1;2)$

9000005810

Level:

Consider the functions $f(x) = x + 1$ and $g(x) = ax + 7$. For what values of the real parameter $a\in \mathbb{R}$, do the functions have the same function value equal to $3$.

$a = -2$

$a = -7$

$a = -1$

$a = 7$

9000007105

Level:

Consider a family $M$ of quadratic functions, as shown in the picture. Any quadratic function in this family is given by the analytic expression \[ y = ax^{2} + bx + c \] where $a$, $b$ and $c$ are real constants and $a\not = 0$. For each function the set $K$ denotes the set of $x$-intercepts. Complete the statement. „The analytic expressions for functions in the family $M$ share only ....”

the solution set $K$

the value of the coefficient $a$

the value of the coefficient $b$

the value of the coefficient $c$

C

9000007604

9000007210

9000007605

9000007608

9000007609

9000007610

9000008008

9000007202

9000005810

9000007105

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