the function f(x)=6x+9/3x-7 is one to one. find its inverse and check your answer.
f^-1(x)=?
please show work
http://www.jiskha.com/display.cgi?id=1346885583
To find the inverse of the function f(x) = (6x + 9) / (3x - 7), we can follow these steps:
Step 1: Replace f(x) with y.
y = (6x + 9) / (3x - 7)
Step 2: Swap x and y.
x = (6y + 9) / (3y - 7)
Step 3: Solve the equation for y.
Multiply both sides of the equation by (3y - 7) to eliminate the denominator:
x(3y - 7) = 6y + 9
Distribute the x to both terms:
3xy - 7x = 6y + 9
Move all terms involving y to one side:
3xy - 6y = 7x + 9
Factor out y on the left side:
y(3x - 6) = 7x + 9
Divide both sides of the equation by (3x - 6) to isolate y:
y = (7x + 9) / (3x - 6)
We have obtained the inverse function as f^(-1)(x) = (7x + 9) / (3x - 6).
To check if this inverse function is correct, we can verify if applying the inverse to f(x) will yield x.
Let's substitute f^(-1)(x) into the original function f(x):
f(f^(-1)(x)) = f((7x + 9) / (3x - 6))
Now simplify the expression:
f(f^(-1)(x)) = [(6[(7x + 9) / (3x - 6)]) + 9] / [3[(7x + 9) / (3x - 6)] - 7]
Multiply both the numerator and denominator on the left side by (3x - 6), and simplify the expression:
f(f^(-1)(x)) = [6(7x + 9) + 9(3x - 6)] / [3(7x + 9) - 7(3x - 6)]
Expand and collect the like terms:
f(f^(-1)(x)) = (42x + 54 + 27x - 54) / (21x + 27 - 21x + 42)
Simplify further:
f(f^(-1)(x)) = (69x) / (69)
f(f^(-1)(x)) = x
As the result is x, we have confirmed that f^(-1)(x) is indeed the inverse function of f(x).