To determine h^(-1)(x), which represents the inverse function of h(x), you need to first find the inverse of the function h(x). The function h(x) in your equation is:
h(x) = 3x - 1 / (x + 5)
To find the inverse, switch the roles of x and y and solve for y:
x = 3y - 1 / (y + 5)
Now, solve for y:
1. Multiply both sides by (y + 5) to get rid of the fraction:
x(y + 5) = 3y - 1
2. Distribute x on the left side:
xy + 5x = 3y - 1
3. Move the term with y to one side by subtracting xy and adding 1 to both sides:
5x + 1 = 3y - xy
4. Now, isolate y:
3y - xy = 5x + 1
3y = 5x + 1 + xy
y = (5x + 1 + xy) / 3
So, the inverse function h^(-1)(x) is:
h^(-1)(x) = (5x + 1 + xy) / 3
Now, you can use this inverse function to solve the equation:
3x - 1 / (x + 5) = 2
First, replace h(x) with its inverse h^(-1)(x):
h^(-1)(x) = (5x + 1 + xy) / 3
Now, replace h(x) with x:
(5x + 1 + xy) / 3 = 2
Now, solve for x:
5x + 1 + xy = 6
Subtract 1 from both sides:
5x + xy = 5
Factor out x:
x(5 + y) = 5
Now, divide by (5 + y):
x = 5 / (5 + y)
This is the solution for x in terms of y. If you have a specific value of y, you can find the corresponding x value.