Since the function is linear, it can be written in slope intercept form which is expressed as
y = mx + c
where
m = slope
c = y intercept
The formula for calculating slope is expressed as
m = (y2 - y1)/(x2 - x1)
From the information given,
g(2) = 7
This means that if x2 = 2, y2 = 7
Also,
g(- 2) = - 1
This means that if x1 = - 2, y1 = - 1
By substituting these values into the formula for calculating slope, we have
m = (7 - - 1)/(2 - - 2) = (7 + 1)/(2 + 2) = 8/4 = 2
We would find the y intercept, c by substituting x = 2, y = 7 and m = 2 into the slope intercept equation. We have
7 = 2 * 2 + c
7 = 4 + c
c = 7 - 4 = 3
By substituting m = 2 and c = 3 into the slope intercept equation, the linear function is
y = 2x + 3
Writing it as a function in terms of g, it is
g(x) = 2x + 3
To find g(- 5), we would substitue x = - 5 into g(x) = 2x + 3, we have
g(- 5) = 2(- 5) + 3 = - 10 + 3
g(- 5) = - 7