9000007608
Level:
C
Find the range of the function \(f(x) = 1 + \left | \frac{1}
{2(x-2)}\right |\).
\((1;\infty )\)
\(\mathbb{R}\)
\(\mathbb{R}\setminus \{1\}\)
\((0;\infty )\)
\(\mathbb{R}\setminus \{2\}\)
C
9000007609
Level:
C
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:
C
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:
C
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:
C
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 \(x\).
\(a = 0\),
\(b = 4\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
\(a = 0\),
\(b = 3\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
\(a = 3\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
\(a = -3\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
9000007203
Level:
C
Consider the function
\[
f(x) =\mathop{ \mathrm{sgn}}\nolimits (x - 2)
\]
defined on \(\mathop{\mathrm{Dom}}(f) =\mathbb{R} ^{-}\). Find
the parameters \(a\)
a \(b\) and
domain of the linear function
\[
g(x) = ax + b
\]
which ensure that \(f\)
and \(g\)
are identical functions.
\[ \]
Hint: The function \(y =\mathop{ \mathrm{sgn}}\nolimits (x)\)
is the sign function. The values of sign function is
\(1\) for each
positive \(x\),
\(- 1\) for each
negative \(x\)
and \(0\) if
\(x = 0\).
\(a = 0\),
\(b = -1\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = 0\),
\(b = 1\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)
\(a = 1\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = -1\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)
9000007207
Level:
C
Identify a function which has the following three properties: it has at least one minimum or maximum, it is an increasing function and the range of this function is the set of all nonnegative numbers.
\(f(x) = 2x - 2\),
\(x\in [ 1;+\infty )\)
\(f(x) = 2x + 2\),
\(x\in (-1;+\infty )\)
\(f(x) = -2x + 2\),
\(x\in (-\infty ;1] \)
\(f(x) = -2x - 2\),
\(x\in \mathbb{R}\)
9000007208
Level:
C
Paul's home is \(6\, \mathrm{km}\) from
the school. At the time \(t = 0\)
Paul starts to walk from his home to the school along a straight street at a constant
velocity \(5\, \mathrm{km}/\mathrm{h}\).
Find the function which describes Paul's remaining distance to the school as a
function of time.
\(s = 6 - 5t\)
\(s = 5t - 6\)
\(s = 5t\)
\(s = 5t + 6\)
9000007209
Level:
C
A current-voltage characteristics is shown in the graph. Find the current
\(I\) as a function of
the voltage \(U\).
\(I = \frac{2}
{3}U -\frac{4}{3};U\in [2;\infty)
\)
\(I = \frac{3}
{2}U - 2;U\in [2;\infty)
\)
\(I = \frac{3}
{2}U + 2;U\in [2;\infty)
\)
\(I = \frac{2}
{3}U + 2;U\in [2;\infty)
\)
9000007809
Level:
C
The price of a goods in a shop is \(\$15\)
per item. The Internet price in an e-shop is cheaper by
\(\$2\) per item. The shipping
cost of the e-shop is \(\$125\).
What is the minimal number of items, which makes the total cost for a transaction
smaller in the e-shop?
\(63\)
\(9\)
\(62\)
\(125\)
\(126\)