To answer this question, we have that both functions are linear functions defined in some intervals. We can find the line equations for those lines as follows:
Finding function f in the interval [-2, -1]
1. We need to define the function f using the points:
(-2, 2) and (-1, -1).
Using these points, we can find the line equation using the two-point form of the line equation:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
(-2, 2) ---> x1 = -2, y1 = 2
(-1, -1) ---> x2 = -1, y2 = -1
[tex]y-2=\frac{-1-2}{-1-(-2)}(x-(-2))\Rightarrow y-2=\frac{-3}{-1+2}(x+2)[/tex][tex]y-2=\frac{-3}{1}(x+2)\Rightarrow y-2=-3(x+2)=-3x-6_{}[/tex][tex]y-2=-3x-6\Rightarrow y=-3x-6+2\Rightarrow y=-3x-4[/tex][tex]f(x)=-3x-4[/tex]
Therefore, the function f(x) = -3x - 4 in the interval [-2, -1]
Finding the function g in the interval [-1, 0]
To find the function g, we can proceed in a similar way:
1. We have the following points:
(-1, 4) and (0, -2)
Then, we have:
(-1, 4) ---> x1 = -1, y1 = 4
(0, -2) ---> x2 = 0, y2 = -2
2. Applying the two-point form of the line, we have:
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)\Rightarrow y-4=\frac{-2-4}{0-(-1)}(x-(-1))[/tex][tex]y-4=\frac{-6}{1}(x+1)\Rightarrow y-4=-6x-6[/tex][tex]y=-6x-6+4\Rightarrow y=-6x-2\Rightarrow g(x)=-6x-2[/tex]
Therefore, the function g(x) = -6x - 2 in the interval [-1, 0].
If we use the given options in the question, we have that:
[tex]g(x)=2f(x-1)[/tex]
We have that:
[tex]2f(x-1)\Rightarrow f(x-1)\Rightarrow f(x-1)=-3(x-1)-4_{}[/tex]
Then, we have:
[tex]f(x-1)=-3x+3-4=-3x-1[/tex]
Then
[tex]2f(x-1)=2(-3x-1)=-6x-2[/tex]
Therefore
[tex]2f(x-1)=-6x-2=g(x)\Rightarrow g(x)=2f(x-1)[/tex]
