Determine the inverse of the function f(x)= (3x+5)/(3x-5).
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f(x) = (3x+5)/(3x-5)
To find the inversem, let f(x) = y
y = (3x+5)/(3x-5)
Cross multiply:
y(3x-5) = 3x + 5
3xy - 5y = 3x + 5
3xy - 3x = 5y + 5
factor out x:
x (3y - 3) = 5y + 5
divide by 3y-3
==> x = (5y+5)/(3y-3)
Then the inverse is:
==> f(x)^-1 = (5x+5)/(3x-3)
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f(x) = (3x+5)/(3x-5). To find the inverse.
We shall find the inverse by solving for x and we see that f(x) o doesnot exist for 3x = 5.
f(x) = y =( 3x+5)/(3x-5)
Multiply by 3x-5. And let us try to make x subject.
y(3x-5) = 3x+5
Collect x's together.
y*3x-3x = 5+5y
3x(y-1) =5(y+1)
x = 5(y+1)/{3(y-1)} . Swapping, x and y,
y = 5(x+1)/{3(x-1)} is the inverse.
We'll re-write the function in a convenient way:
f(x)=(3x+5)/(3x-5) as y=(3x+5)/(3x-5)
Now, we'll solve this equation for x, multiplying both sides by (3x-5):
3xy-5y = (3x+5)
We'll move all terms containing x, to the left side and all terms in y, to the right side:
3xy-3x = 5+5y
We'll factorize:
x(3y-3) = 5(1+y)
x= 5(1+y) / (3y-3)
Now, we'll interchange x and y:
y = 5(1+x) / (3x-3)
So, the inverse function is:
[f(x)]^(-1) = 5(1+x) / (3x-3)
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