the function f(x)=6x+9/3x-7
find the inverse and check answer
f^-1(x)=?
please show work
To find the inverse of a function, we need to swap the roles of x and y and solve for y.
Given the function f(x) = (6x + 9)/(3x - 7), we'll swap the roles of x and y:
x = (6y + 9)/(3y - 7)
Now, let's solve for y:
Multiply both sides of the equation by (3y - 7) to eliminate the denominator:
x(3y - 7) = 6y + 9
Distribute x on the left side:
3xy - 7x = 6y + 9
Rearrange the equation to isolate y terms on one side:
3xy - 6y = 7x + 9
Factor out y:
y(3x - 6) = 7x + 9
Divide both sides of the equation by (3x - 6):
y = (7x + 9)/(3x - 6)
So, the inverse function of f(x) = (6x + 9)/(3x - 7) is f^(-1)(x) = (7x + 9)/(3x - 6).
To check the answer, substitute the inverse function into the original function and see if we get x back.
Original function: f(x) = (6x + 9)/(3x - 7)
Inverse function: f^(-1)(x) = (7x + 9)/(3x - 6)
Substituting the inverse function into the original function:
f(f^(-1)(x)) = (6[(7x + 9)/(3x - 6)] + 9)/(3[(7x + 9)/(3x - 6)] - 7)
Simplify the expression:
f(f^(-1)(x)) = (42x + 54)/(7x + 9 - 7) = (42x + 54)/(7x + 9)
Simplifying further:
f(f^(-1)(x)) = (6x + 9)/(1) = 6x + 9
As we can see, f(f^(-1)(x)) simplifies back to the original input x. Therefore, our inverse function is correct.