C

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

9000007207

Level: 
C
Identify a function which has the following three properties: it has at least one minimum or maximum, it is an increasing function and the range of this function is the set of all nonnegative numbers.
\(f(x) = 2x - 2\), \(x\in [ 1,+\infty )\)
\(f(x) = 2x + 2\), \(x\in (-1,+\infty )\)
\(f(x) = -2x + 2\), \(x\in (-\infty ,1] \)
\(f(x) = -2x - 2\), \(x\in \mathbb{R}\)

9000007209

Level: 
C
A current-voltage characteristics is shown in the graph. Find the current \(I\) as a function of the voltage \(U\).
\(I = \frac{2} {3}U -\frac{4}{3},U\in [2,\infty) \)
\(I = \frac{3} {2}U - 2,U\in [2,\infty) \)
\(I = \frac{3} {2}U + 2,U\in [2,\infty) \)
\(I = \frac{2} {3}U + 2,U\in [2,\infty) \)

9000007101

Level: 
C
Consider a family \(M\) of quadratic functions, as shown in the picture. Any quadratic function in this family is given by the analytic expression \[ y = ax^{2} + bx + c \] where \(a\), \(b\) and \(c\) are real constants and \(a\not = 0\). For each function the set \(K\) denotes the set of \(x\)-intercepts. Complete the statement. „The analytic expressions for functions in the family \(M\) differ only in ....”
the coefficient \(a\)
the coefficient \(b\)
the coefficient \(c\)
the set \(K\)

9000007201

Level: 
C
Consider the function \[ f(x) = [x + 2] \] defined on the domain \(\mathop{\mathrm{Dom}}(f) = (1,2)\). Find the parameters \(a\) and \(b\) in the linear function \[ g(x) = ax + b \] which ensure that the functions \(f\) and \(g\) are identical on the domain of \(f\). \[ \] 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 \(x\).
\(a = 0\), \(b = 3\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)
\(a = 3\), \(b = 0\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)
\(a = 0\), \(b = 4\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)
\(a = -3\), \(b = 0\), \(\mathop{\mathrm{Dom}}(g) = (1,2)\)

9000004905

Level: 
C
In the following list identify a statement which is not true for the function \(f\colon y = |\log (x - 3) - 1|\).
The function \(f\) is increasing on the domain.
The domain of the function \(f\) is \((3,\infty )\).
All values of the function \(f\) are nonnegative.
The function \(f\) does not have a \(y\)-intercept.
The \(x\)-intercept of the function \(f\) is \(x = 13\).
The function \(f\) is not one-to-one.