Answer:
Step-by-step explanation:
As first step we must obtain the equation of the line. According to Analytical Geometry, we can get a equation of the line by knowing two different points. A linear function is represented by the following expression:
[tex]y = m\cdot x + b[/tex]
Where:
[tex]x[/tex] - Independent variable, dimensionless.
[tex]y[/tex] - Dependent variable, dimensionless.
[tex]m[/tex] - Slope, dimensionless.
[tex]b[/tex] - y-Intercept, dimensionless.
If we know that [tex](x_{1}, y_{1}) = (0, 5)[/tex] and [tex](x_{2}, y_{2}) = (3, 8)[/tex], the following system of linear equations is formed:
[tex]5 = 0\cdot m + b[/tex]
[tex]b = 5[/tex] (Eq. 1)
[tex]8 = 3\cdot m + b[/tex]
[tex]3\cdot m + b = 8[/tex] (Eq. 2)
From (Eq. 1) in (Eq. 2) we get the value of [tex]m[/tex]:
[tex]3\cdot m + 5 = 8[/tex]
[tex]3\cdot m = 3[/tex]
[tex]m = 1[/tex]
The equation of the line that contains the points (0, 5) and (3, 8) is [tex]f(x) = x + 5[/tex].
As next step we must apply a reflection in the x-axis, whose operation is defined as follows:
[tex](x', y') \longrightarrow (x, -y)[/tex]
That is:
[tex]x' = x[/tex]
[tex]y' = -y[/tex]
Then, the function [tex]g(x)[/tex] is [tex]g(x) = -x-5[/tex]. We plot each function and include the result in the attachment below.
