Rational functions

1103129201

Level: 
B
The thin lens equation \( \frac1a+\frac1{a'}=\frac1f \) describes the quantitative relationship between the object distance \( a \), the image distance \( a' \), and the focal length \( f \). Let the focal length of a thin lens be \( 0{.}5\,\mathrm{m} \). Choose the picture that shows the object distance vs. image distance graph, if \( a\in [0.1\,\mathrm{m};0.4\,\mathrm{m}]\cup [0.6\,\mathrm{m};3.0\,\mathrm{m}] \).

2000006701

Level: 
B
A part of the graph of the function \( f(x)=-\frac2x \) is shown in the picture. Identify which of the following statements is true.
The function \( g \) defined by \( g(x)=-\left|f(x)\right| \) is an even function.
The function \( g \) defined by \( g(x)=-\left|f(x)\right| \) is bounded below.
The function $f$ is decreasing on \( (-\infty;0)\).
The function $m$ defined by \( m(x)=f(x)-3 \) is bounded.

2000006704

Level: 
B
Let \(X\) and \(Y \) be intersections of the graph of the function \(f(x) = \frac{3x-5} {2+x}\) with \(x\)- and \(y\)-axis, respectively. Find these points.
\(X = \left[\frac{5}{3};0\right]\), \(Y = \left[0;-\frac{5}{2}\right]\)
\(X = \left[-\frac{5}{2};0\right]\), \(Y = \left[0;\frac{5}{3}\right]\)
\(X = \left[0;\frac{5}{3}\right]\), \(Y = \left[-\frac{5}{2};0\right]\)
\(X = \left[\frac{5}{2};0\right]\), \(Y = \left[0;-\frac{5}{3}\right]\)

2010009901

Level: 
B
Find the domain \(\mathrm{Dom}(f)\) and range \(\mathop{\mathrm{Ran}}(f)\) of the function \(f(x) = \frac{x-3} {x+1}\).
\begin{align*} \mathrm{Dom}(f) &= (-\infty ;-1)\cup (-1;\infty ),\\ \mathop{\mathrm{Ran}}(f) &= (-\infty ;1)\cup (1;\infty ) \end{align*}
\begin{align*} \mathrm{Dom}(f) &= (-\infty ;1)\cup (1;\infty ),\\ \mathop{\mathrm{Ran}}(f) &= (-\infty ;-1)\cup (-1;\infty ) \end{align*}
\begin{align*} \mathrm{Dom}(f) &= (-\infty ;3)\cup (3;\infty ),\\ \mathop{\mathrm{Ran}}(f) &= (-\infty ;-1)\cup (-1;\infty ) \end{align*}
\begin{align*} \mathrm{Dom}(f) &= (-\infty ;-3)\cup (-3;\infty ),\\ \mathop{\mathrm{Ran}}(f) &= (-\infty ;1)\cup (1;\infty ) \end{align*}

2010015101

Level: 
B
Let by \(X\) and \(Y\) denote the intersection points of the graph of the function \(f(x)=\frac{2}{x+3}-1\) with \(x\) and \(y\)-axis, respectively. Find coordinates of \(X\) and \(Y\).
\(X = [-1;0]\), \(Y = \left[0;-\frac13\right]\)
\(X = [1;0]\), \(Y = \left[0;\frac13\right]\)
\(X = \left[-\frac13;0\right]\), \(Y = [0;-1]\)
\(X = [-3;0]\), \(Y = [0;-1]\)