Answer:
(x^2 +6)/2
or 1/2 x^2 +3
Step-by-step explanation:
f(x)= sqrt(2x-6)
y =sqrt(2x-6)
To find the inverse function, we interchange the x and y and solve for y
x = sqrt(2y-6)
Square each side
x^2 = sqrt(2y-6)^2
x^2 = 2y-6
Add 6 to each to each side
x^2 +6 = 2y-6+6
x^2 +6 = 2y
Divide each side by 2
(x^2 +6)/2 = 2y/2
(x^2 +6)/2 = y
The inverse function is
(x^2 +6)/2
or 1/2 x^2 +3
Two functions f(x) and g(x) are inverses if:
f(g(x)) = x = g(f(x))
The solution is:
[tex]g(x) = \frac{1}{2} x^{2} + 3[/tex]
Now we want to find the inverse function to:
[tex]f(x) = \sqrt{2x - 6}[/tex]
Because this is a square root function, we know that the inverse must be a quadratic function, so let's try with something like:
[tex]g(x) = a*x^2 + c[/tex]
Now we can use the first property to get:
[tex]g(f(x)) = a*f(x)^2 + c = x\\\\ = a*\sqrt{2x - 6}^2 + c = x\\\\= a*(2x - 6) + c = x\\\\2*a*x -6*a + c = x[/tex]
Then we must have:
2*a =1
a = 1/2
And:
-6*a + c = 0
-6*(1/2) + c = 0
-3 + c = 0
c = 4
Then the inverse function is:
[tex]g(x) = \frac{1}{2} x^{2} + 3[/tex]
If you want to learn more, you can read:
https://brainly.com/question/10300045
