B2. The family of functions is given by $f(x)=3^{x-b}-\frac{1}{3}, b \in \mathbb{R}$.

a) For $b=-1$, draw the graph of the function $f$ and write down its range of values.

b) Calculate the value of the parameter $b$ for which $x=\frac{1}{2}$ is a zero of the function $f$.

c) For $b=2$, calculate to two decimal places the abscissa of the intersection point of the graph of the function $f$ with the line $y-6=0$.
B2.

a) We insert $b=-1 \mathrm{v}$ into the function $f$ and draw its graph: the graph can be drawn with shifts along the abscissa and ordinate axes or by calculating the zeros, the intersection with the ordinate axis, and the horizontal asymptote. We also write down its range of values $\left(-\frac{1}{3}, \infty\right)$.

b) We insert $x=\frac{1}{2} \mathrm{v}$ into the function $f$ and obtain the equation $3^{\frac{1}{2}-b}-\frac{1}{3}=0$. Solving the equation, we get the solution $b=\frac{3}{2}$.

c) We insert $b=2$ into the function $f$ and set it equal to 6, transforming the equation to the form $3^{x-2}=\frac{19}{3}$. We solve it by logarithmizing and thus calculate the abscissa of the intersection point to two decimal places $x \doteq 3.7$.

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=71&width=1762&top_left_y=1592&top_left_x=158)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=66&width=1753&top_left_y=1646&top_left_x=163)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=74&width=1756&top_left_y=1696&top_left_x=161)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=65&width=1751&top_left_y=1755&top_left_x=164)

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=62&width=1756&top_left_y=1805&top_left_x=161)

Solving the equation (logarithmizing) ..................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_5bf9d67aee417dc7ba6ag-23.jpg?height=60&width=1756&top_left_y=1906&top_left_x=161)