9000014208
Level:
B
Identify a possible analytic expression for the function
\(f\)
graphed in the picture.
\(f(x) = \frac{2x+1}
{x-1} \)
\(f(x) = \frac{3}
{x-1} - 2\)
\(f(x)= \frac{2x-1}
{x+1} \)
\(f(x) = \frac{2x+2}
{x-1} \)
Rational functions
9000014206
Level:
B
Find the domain \(\mathrm{Dom}(f)\)
and range \(\mathop{\mathrm{Ran}}(f)\) of
the function \(f(x) = \frac{2+x}
{x+4}\).
\begin{align*}
\mathrm{Dom}(f) &= (-\infty ;-4)\cup (-4;\infty ),\\
\mathop{\mathrm{Ran}}(f) &= (-\infty ;1)\cup (1;\infty )
\end{align*}
\begin{align*}
\mathrm{Dom}(f) &= (-\infty ;4)\cup (4;\infty ), \\
\mathop{\mathrm{Ran}}(f) &= (-\infty ;1)\cup (1;\infty )
\end{align*}
\begin{align*}
\mathrm{Dom}(f) &= (-\infty ;2)\cup (2;\infty ),\\
\mathop{\mathrm{Ran}}(f) &= (-\infty ;4)\cup (4;\infty )
\end{align*}
\begin{align*}
\mathrm{Dom}(f) &= (-\infty ;4)\cup (4;\infty ), \\
\mathop{\mathrm{Ran}}(f) &= (-\infty ;2)\cup (2;\infty )
\end{align*}
9000014209
Level:
B
Consider the function \(f(x) = \frac{3x+1}
{x-2} \).
Find all \(x\)
such that \(f(x) > 0\).
\(x\in \left (-\infty ;-\frac{1}
{3}\right )\cup (2;\infty )\)
\(x\in \left (-\frac{1}
{3};\infty \right )\)
\(x\in (2;3)\)
\(x\in (-\infty ;-3)\cup (2;\infty )\)
9000014210
Level:
B
Consider the function \(f(x) = \frac{2x+1}
{x+3} \).
Find all \(x\)
such that \(f(x) < 0\).
\(x\in \left (-3;-\frac{1}
{2}\right )\)
\(x\in (-\infty ;-3)\cup \left (\frac{1}
{2};\infty \right )\)
\(x\in (-3;\infty )\)
\(x\in \left (-\infty ;-\frac{1}
{2}\right )\)
9000014207
Level:
B
Identify a possible analytic expression for the function
\(f\)
graphed in the picture.
\(f(x)= \frac{x+1}
{x+2}\)
\(f(x)= \frac{1}
{x+2} - 1\)
\(f(x) = \frac{1}
{x+2} + 1\)
\(f(x) = \frac{x+1}
{x-2}\)
9000009908
Level:
A
Consider a function
\[
f(x) = \frac{-3}
{x}
\]
defined on the domain \(\mathrm{Dom}(f) =\mathbb{R}\setminus \{ - 1.0\}\).
Find the range of this function.
\(\mathbb{R}\setminus \{0.3\}\)
\(\mathbb{R}\setminus \{0\}\)
\(\mathbb{R}\setminus \{ - 3.0\}\)
\(\mathbb{R}\)
9000009904
Level:
C
Consider the functions
\[
\text{$f(x)= \frac{2}
{x}$ and $g(x) = \frac{2x}
{x^{2}} $.}
\]
Is the conclusion \(f = g\)
valid?
Yes.
No.
Yes, but only on \(\mathbb{R}^{+}\).
Yes, but only on \(\mathbb{R}^{-}\).
9000009906
Level:
C
Consider a function
\[
f(x) = \frac{k}
{x}
\]
with a nonzero real parameter \(k\).
Describe what happens with the function
\(f\) if the
coefficient \(k\)
changes the sign.
The function changes the type of monotonicity on the sets
\(\mathbb{R}^{+}\) and
\(\mathbb{R}^{-}\)
(either from an increasing function into a decreasing function or vice versa).
The function changes its parity (from an odd function into an even function or from
an even function into an odd function).
The domain of the function changes.
None of the above, both functions have the same parity, monotonicity and domain.
9000009907
Level:
C
Consider a function
\[
f(x) = \frac{k}
{x}
\]
with a nonzero real parameter \(k\).
Suppose that the value of the coefficient
\(k\) changes, but
the sign of \(k\)
remains the same. Describe which of the properties of
\(f\) is
changed.
None of the above, both functions have the same parity, monotonicity and range.
The function changes its parity (from an odd function into an even function or from
en even function into an odd function).
The range of the function changes.
The function changes the type of monotonicity on the sets
\(\mathbb{R}^{+}\) and
\(\mathbb{R}^{-}\)
(either from an increasing function into a decreasing function or vice versa).
9000009910
Level:
A
A body is deformed continuously in a machine press. The density
\(\rho \) is inversely proportional
to the volume \(V \) of the body,
i.e. there exists a constant \(k\)
such that
\[
\rho = \frac{k}
{V }.
\]
Find the constant \(k\)
(including the correct unit) if it is known that the density was
\(\rho = 25\: \frac{\mathrm{kg}}
{\mathrm{m}^{3}} \) when the body
had volume \(V = 2\, \mathrm{dm}^{3}\).
\(50\, \mathrm{g}\)
\(12.5\, \mathrm{g}\)
\(12.5\, \mathrm{m}\)
\(50\, \mathrm{m}\)