Rational functions

9000025808

Level: 
C
In the following list identify a true statement on the function \(f\). \[ f(x) = \frac{(x - 1)(x + 2)} {(2x + 1)(3 - 2x)} \]
\(f(x) > 0 \iff x\in \left (-2;-\frac{1} {2}\right )\cup \left (1; \frac{3} {2}\right )\)
\(f(x) > 0 \iff x\in (-\infty ;-2)\cup \left (-\frac{1} {2};1\right )\cup \left (\frac{3} {2};\infty \right )\)
\(f(x) > 0 \iff x\in (-\infty ;-2)\cup (1;\infty )\)
\(f(x) > 0 \iff x\in \left (-2; \frac{3} {2}\right )\)

9000025809

Level: 
C
In the following list identify a true statement on the function \(f\). \[ f(x)= \frac{(6x - 1)} {(x - 2)(3x + 1)} \]
\(f(x)\geq 0 \iff x\in \left (-\frac{1} {3}; \frac{1} {6}\right ] \cup (2;\infty )\)
\(f(x)\geq 0 \iff x\in \left (-\frac{1} {3}; \frac{1} {6}\right )\cup (2;\infty )\)
\(f(x)\geq 0 \iff x\in \left (-\infty ;-\frac{1} {3}\right )\cup \left [ \frac{1} {6};2\right )\)
\(f(x)\geq 0 \iff x\in \left [ -\frac{1} {3}; \frac{1} {6}\right ] \cup (2;\infty )\)

9000025810

Level: 
C
In the following list identify a true statement on the function \(f\). \[ f(x) = \frac{(x - 2)(3 - x)} {(2x - 1)(3x - 1)} \]
\(f(x)\geq 0 \iff x\in \left (\frac{1} {3}; \frac{1} {2}\right )\cup [ 2;3] \)
\(f(x)\geq 0 \iff x\in \left [ \frac{1} {3}; \frac{1} {2}\right ] \cup [ 2;3] \)
\(f(x)\geq 0 \iff x\in \left (-\infty ; \frac{1} {3}\right )\cup \left [ \frac{1} {2};2\right ] \cup [ 3;\infty )\)
\(f(x)\geq 0 \iff x\in \left (\frac{1} {3}; \frac{1} {2}\right )\cup (2;3)\)

9000025803

Level: 
C
Find all intersections of the graph of the following function with \(x\)-axis. \[ f(x) = \frac{2x + 1} {x^{2} - x - 6} \]
\(X = \left [-\frac{1} {2};0\right ]\)
\(X = \left [-\frac{1} {6};0\right ]\)
\(X_{1} = [-2;0]\), \(X_{2} = [3;0]\)
\(X_{1} = [-2;0]\), \(X_{2} = \left [-\frac{1} {2};0\right ]\), \(X_{3} = [3;0]\)

9000025806

Level: 
C
In the following list identify a true statement on the function \(f\). \[ f(x)= \frac{(3x - 1)(2 - x)} {x + 2} \]
\(f(x) > 0 \iff x\in (-\infty ;-2)\cup \left (\frac{1} {3};2\right )\)
\(f(x) > 0 \iff x\in \left (-2; \frac{1} {3}\right )\cup (2;\infty )\)
\(f(x) > 0 \iff x\in \left (-\infty ; \frac{1} {3}\right )\cup (2;\infty )\)
\(f(x) > 0 \iff x\in \left (-\infty ; \frac{1} {3}\right )\)

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*}