Answer:
[tex]\displaystyle f(x) = -\frac{2}{3}\, x - \frac{4}{3}[/tex].
Step-by-step explanation:
Rewrite the slope of the given line by rewriting to equation of that line into the slope-intercept form [tex]y = m_{1} \, x + b[/tex]:
[tex]\displaystyle y = \frac{3}{2}\, x + (-2)[/tex].
Thus, the slope of the given line would be [tex]m_{1} = (3/2)[/tex].
Let [tex]m_{2}[/tex] denote the slope of the line in question. Two lines in a plane are perpendicular to one another if and only if the product of their slopes is [tex](-1)[/tex]. Thus, [tex]m_{1}\, m_{2} = (-1)[/tex] since these two lines are perpendicular to one another.
Since [tex]m_{1} = (3/2)[/tex], the slope of the line in question would be:
[tex]\begin{aligned}m_{2} &= \frac{(-1)}{m_{1}} \\ &= \frac{(-1)}{(3/2)} \\ &= -\frac{2}{3}\end{aligned}[/tex].
Let [tex]b[/tex] denote the [tex]y[/tex]-intercept of the line in question (the value of this [tex]b\![/tex] needs to be found.) The slope-intercept equation of the line in question would be:
[tex]\displaystyle y = \left(-\frac{2}{3}\right) \, x + b[/tex].
In other words, for a point on this line, if the input ([tex]x\![/tex]-coordinate) is [tex]x[/tex], the output ([tex]y[/tex]-coordinate) would be [tex]((-2/3)\, x + b)[/tex]. This relation is equivalent to the function:
[tex]\displaystyle f(x) = -\frac{2}{3} \, x + b[/tex].
The question stated that [tex]f(1) = -2[/tex]. In other words, [tex]((-2/3)\, x + b)[/tex] should evaluate to [tex](-2)[/tex] when the input variable [tex]x [/tex] is replaced with [tex]1[/tex]. Thus:
[tex]\displaystyle -\frac{2}{3} + b = -2[/tex].
Solve for [tex]b[/tex]:
[tex]\displaystyle b = -\frac{4}{3}[/tex].
Therefore, the function that represents this line would be:
[tex]\displaystyle f(x) = -\frac{2}{3}\, x - \frac{4}{3}[/tex].
Here's another example on how to find the equation of a line perpendicular to a given line: https://brainly.com/question/26231200.
