the function f(x)=6x+9/3x-7 find inverse and check it
f^-1(x)=?
please show work
To find the inverse of the given function, f(x) = (6x + 9)/(3x - 7), we need to follow these steps:
Step 1: Replace f(x) with y to obtain the equation in terms of y.
y = (6x + 9)/(3x - 7)
Step 2: Swap x and y, and then solve for y.
x = (6y + 9)/(3y - 7)
Step 3: Multiply both sides of the equation by (3y - 7) to eliminate the denominator.
x(3y - 7) = 6y + 9
Step 4: Expand the equation.
3xy - 7x = 6y + 9
Step 5: Move the terms involving y to one side of the equation.
3xy - 6y = 7x + 9
Step 6: Factor out y on the left side.
y(3x - 6) = 7x + 9
Step 7: Divide both sides of the equation by (3x - 6) to solve for y.
y = (7x + 9)/(3x - 6)
Thus, the inverse of the given function is: f^(-1)(x) = (7x + 9)/(3x - 6).
To check if this is the correct inverse, you can verify whether composing the original function with its inverse gives the identity function. In this case, we would check if f(f^(-1)(x)) = x.
Let's perform the composition:
f(f^(-1)(x)) = f((7x + 9)/(3x - 6))
Plugging this function into the original equation for f(x) gives:
f(f^(-1)(x)) = (6((7x + 9)/(3x - 6)) + 9)/(3((7x + 9)/(3x - 6)) - 7)
Simplifying this expression, we can check if it equals x. If it does, then (7x + 9)/(3x - 6) is indeed the inverse of the original function.