Two functions f(x) and g(x) are inverses if and only if:
f(g(x)) = x
g(f(x)) = x
We will find that the anwer here is g(x) = -2*x - 3
We want to find the inverse function of
[tex]f(x) = -\frac{1}{2} x - \frac{3}{2}[/tex]
Because f(x) is a linear function, its inverse is also a linear function, so at the moment we can write:
[tex]g(x) = a*x + b[/tex]
The composition will be:
[tex]g(f(x)) = a*(-\frac{1}{2}x - \frac{3}{2} ) + b = x[/tex]
Now we can expand the above expression to find the values of a and b.
[tex]g(f(x)) = a*(-\frac{1}{2}x - \frac{3}{2} ) + b = x\\\\ = -\frac{a}{2}*x - \frac{3*a}{2} + b = x\\\\[/tex]
Then we must have:
[tex]\frac{-a}{2} = 1\\\\a = -2[/tex]
and
[tex]\frac{-3a}{2} + b = 0\\b = \frac{3a}{2} = \frac{3*(-2)}{2} = -3[/tex]
Then the inverse function is:
g(x) = -2*x - 3
If you want to learn more, you can read:
https://brainly.com/question/10300045
