Here are the solutions to the problems: 3. Calculate the sum of terms of the first 12 even numbers of an Arithmetic Progression: 1, 2, 3, 4,.... Step 1: Identify the sequence of even numbers from the given arithmetic progression. The given arithmetic progression is $1, 2, 3, 4, \dots$. The even numbers in this sequence are $2, 4, 6, 8, \dots$. This is an arithmetic progression with: First term, $a = 2$ Common difference, $d = 4 - 2 = 2$ Number of terms, $n = 12$ Step 2: Use the formula for the sum of an arithmetic progression. The sum of the first $n$ terms of an arithmetic progression is given by: $$S_n = \frac{n}{2}(2a + (n-1)d)$$ Substitute the values: $a=2$, $d=2$, $n=12$. $$S_{12} = \frac{12}{2}(2(2) + (12-1)2)$$ $$S_{12} = 6(4 + (11)2)$$ $$S_{12} = 6(4 + 22)$$ $$S_{12} = 6(26)$$ $$S_{12} = 156$$ The sum of the first 12 even numbers is $\boxed{\text{156}}$. 4. The range of a function $g(x) = 2x - 1$ is $\{-3, -1, 3, 0\}$. Draw an arrow showing the mapping with its domain. Step 1: Determine the domain values corresponding to the given range values. The function is $g(x) = 2x - 1$. The range is the set of output values, $g(x)$. We need to find the input values, $x$, for each value in the range. Let $y = g(x)$. So, $y = 2x - 1$. To find $x$, rearrange the equation: $y + 1 = 2x$ $x = \frac{y+1}{2}$ Step 2: Calculate the domain values for each range value. • For $y = -3$: $x = \frac{-3+1}{2} = \frac{-2}{2} = -1$ • For $y = -1$: $x = \frac{-1+1}{2} = \frac{0}{2} = 0$ • For $y = 3$: $x = \frac{3+1}{2} = \frac{4}{2} = 2$ • For $y = 0$: $x = \frac{0+1}{2} = \frac{1}{2}$ So, the domain of the function is $\{-1, 0, 2, \frac{1}{2}\}$. Step 3: Draw an arrow diagram showing the mapping. Domain: $\{-1, 0, 2, \frac{1}{2}\}$ Range: $\{-3, -1, 3, 0\}$ The mapping is as follows: $$ \begin{array}{rcl} \text{Domain} & & \text{Range} \\ -1 & \longrightarrow & -3 \\ 0 & \longrightarrow & -1 \\ 2 & \longrightarrow & 3 \\ \frac{1}{2} & \longrightarrow & 0 \end{array} $$