Rational functions

9000009901

Level: 
C
The picture shows parts of the graphs of the functions \[ \text{$f(x)= \frac{k_{1}} {x} $ and $g(x) = \frac{k_{2}} {x} $.} \] Find the relationship between \(k_{1}\) and \(k_{2}\)?
\(k_{1} > k_{2}\)
\(k_{1} < k_{2}\)
\(k_{1} = k_{2}\)
No conclusion is possible, more of the above possibilities may occur.

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 \(V = 2\, \mathrm{dm}^{3}\).
\(50\, \mathrm{g}\)
\(12.5\, \mathrm{g}\)
\(12.5\, \mathrm{m}\)
\(50\, \mathrm{m}\)

9000014203

Level: 
B
Which of the statements from the following list is true for the function \(f(x) = -\frac{2} {x} + 1\)?
The function \(f\) is a one-to-one function.
The function \(f\) is an odd function.
The function \(f\) is an increasing function.
The graph of the function \(f\) is a hyperbola with branches in the second and fourth quadrant.

9000014201

Level: 
B
Find intersection points of the graph of the rational function \( f(x) = \frac{2x - 3} {x - 2} \) with \(y\)-axis.
\(Y = \left [0; \frac{3} {2}\right ]\)
\(Y = \left [\frac{3} {2};0\right ]\)
\(Y _{1} = \left [0; \frac{3} {2}\right ]\text{ and }Y _{2} = \left [\frac{3} {2};0\right ]\)
\(Y = \left [2;2\right ]\)