9000007205
Level:
A
Given the linear function
\[
f(x) = -3x + 2
\]
solve the inequality \(f(x)\geq 1\).
\(x\leq \frac{1}
{3}\)
\(x\geq \frac{1}
{3}\)
\(x\leq \frac{2}
{3}\)
\(x\geq 2\)
Linear functions
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
\(x = 0\).
\(a = 0\),
\(b = -1\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = 0\),
\(b = 1\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)
\(a = 1\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = -1\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)
9000007204
Level:
A
Among the following functions identify an odd function
\(f\) which
satisfies \(f(-2) = 4\).
\(f(x) = -2x\)
\(f(x) = -2x + 1\)
\(f(x) = x + 2\)
\(f(x) = 2x + 8\)
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.
\(s = 6 - 5t\)
\(s = 5t - 6\)
\(s = 5t\)
\(s = 5t + 6\)
9000005807
Level:
C
Consider the function \(f(x) = -x + 4\)
and a triangle which has one side on the graph of
\(f\) and
two other sides on the axes. Find the area of this triangle.
\(8\)
\(4\)
\(6\)
\(10\)
9000005809
Level:
C
Consider the functions \(f(x) = x - 1\)
and \(g(x) y = -x + a\). Find the value
of the real parameter \(a\in \mathbb{R}\)
which ensure that the functions have a common value at
\(x = 3\), i.e.
\(f(3) = g(3)\).
\(a = 5\)
\(a = -1\)
\(a = 1\)
\(a = 2\)
9000005701
Level:
A
Given the linear function \(f(x) = 3x - 2\),
evaluate \(f\left (\frac{1}
{6}\right )\).
\(-\frac{3}
{2}\)
\(- 1\)
\(\frac{1}
{6}\)
\(\frac{5}
{2}\)
9000007201
Level:
C
Consider the function
\[
f(x) = [x + 2]
\]
defined on the domain \(\mathop{\mathrm{Dom}}(f) = (1;2)\).
Find the parameters \(a\)
and \(b\)
in the linear function
\[
g(x) = ax + b
\]
which ensure that the functions \(f\)
and \(g\) are identical
on the domain of \(f\).
\[ \]
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 \(x\).
\(a = 0\),
\(b = 3\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
\(a = 3\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
\(a = 0\),
\(b = 4\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)
\(a = -3\),
\(b = 0\);
\(\mathop{\mathrm{Dom}}(g) = (1;2)\)