Answer:
The two functions are inverses.
Step-by-step explanation:
Recall what an inverse is:
When you inverse an equation, like y = 2x for example, you switch, the y and x values. So you'd now have x = 2y, then x/2 = y.
In a case like that we'd say that, y = 2x, and y = x/2 are inverses of each other.
Now looking at your question, all we'd need to do is take the inverse of 1, and see if it matches up with the other one. So pick a function. I'll choose g(x) and I'll inverse it.
[tex]g(x) = y = 2x + 6[/tex]
get rid of the g(x), as it doesn't help you anymore.
[tex]y = 2x + 6[/tex]
then, switch the x and the y.
[tex]x = 2y + 6[/tex]
then isolate for y.
[tex]x - 6 = 2y[/tex]
[tex]\frac{x-6}{2} = y[/tex]
Now let's see, does [tex]f(x) = \frac{1}{2}x - 3[/tex] equal [tex]\frac{x-6}{2}[/tex] ? Well actually yes.
We just didn't simplify all the way.
[tex]\frac{x-6}{2} = y\\[/tex]
divide each term by 2
[tex]\frac{x}{2} - 3 = y[/tex]
[tex]\frac{x}{2}[/tex] can be written as [tex]\frac{1}{2}x[/tex] so we get
[tex]\frac{1}{2}x - 3[/tex] which is equal to f(x)
Therefore the 2 functions are inverses.
Q.E.D
