Linear functions

9000007808

Level: 
B
Given a function \(f(x) = \frac{x} {3} + 1\), find the function \(g\) such that the graph of \(g\) is symmetric with the graph of \(f\) about the \(y\)-axis.
\(g\colon y = -\frac{x} {3} + 1\)
\(g\colon y = 3x + 1\)
\(g\colon y = -3x + 1\)
\(g\colon y = -\frac{x} {3} - 1\)
Such a function does not exist.

9000007809

Level: 
C
The price of a goods in a shop is \(\$15\) per item. The Internet price in an e-shop is cheaper by \(\$2\) per item. The shipping cost of the e-shop is \(\$125\). What is the minimal number of items, which makes the total cost for a transaction smaller in the e-shop?
\(63\)
\(9\)
\(62\)
\(125\)
\(126\)

9000007810

Level: 
C
A fuel tank in a car has the capacity \(40\) litres. The current volume of the fuel in the fuel tank is \(6\) litres. The speed of fuelling is \(1\) litre of gasoline each \(3\) seconds. Find the function which describes the volume of the gasoline in the fuel tank (in litres) as a function of time (in seconds).
\(V = \frac{1} {3}t + 6,\ t\in [ 0;102] \)
\(V = 3t + 6,\ t\in [ 0;102] \)
\(V = 3t + 6,\ t\in [ 0;40] \)
\(V = 3t + 6,\ t\in \mathbb{R}_{0}^{+}\)
\(V = \frac{1} {3}t + 6,\ t\in [ 0;40] \)

9000009301

Level: 
C
An automatic machine produces \(12\) components per minute and stores them in a box with capacity \(1\: 500\) components. The machine starts with an initial amount of \(240\) components in the box. In what time will be the box full?
\(1\, \mathrm{h}\) \(45\, \mathrm{min}\)
\(1\, \mathrm{h}\) \(55\, \mathrm{min}\)
\(2\, \mathrm{h}\) \(5\, \mathrm{min}\)
\(2\, \mathrm{h}\) \(15\, \mathrm{min}\)

9000009302

Level: 
C
An automatic machine produces \(12\) components per minute and stores them in a box with capacity \(1\: 500\) components. The machine starts with an initial amount of \(240\) components in the box. In what time will the box contain \(1\: 020\) components?
\(1\, \mathrm{h}\) \(5\, \mathrm{min}\)
\(55\, \mathrm{min}\)
\(1\, \mathrm{h}\)
\(1\, \mathrm{h}\) \(10\, \mathrm{min}\)

9000007210

Level: 
C
Jane has to reach the opposite shore of a lake. She has three possibilities how to cross. She can use her own boat, start immediately and sail at the velocity \(4\, \mathrm{km}/\mathrm{h}\). Another option is to wait for her friend, Peter, with a faster boat. Peter's boat is capable to sail at the velocity \(10\, \mathrm{km}/\mathrm{h}\). However, his boat will be available in \(1.5\) hours from now. The last option is to use the regular passenger transportation line which will departure in \(2.25\) hours from now and sails at the speed \(20\, \mathrm{km}/\mathrm{h}\). Find the interval of distances to the opposite shore for which the fastest option is to use Peter's boat.
between \(10\) and \(15\) kilometers
up to \(10\) kilometers
between \(15\) and \(20\) kilometers
larger that \(20\) kilometers

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 \(x = 0\).
\(a = 0\), \(b = -1\); \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = 0\), \(b = 1\); \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)
\(a = 1\), \(b = 0\); \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{-}\)
\(a = -1\), \(b = 0\); \(\mathop{\mathrm{Dom}}(g) =\mathbb{R} ^{+}\)