Find the inverse of f(x)=2x+1/x-1 and please VERIFY(CHECK) Thanks so much
y = (2x+1) / (x-1) I assume you mean
x = (2y+1)/(y-1)
x y -x = 2 y + 1
2 y - xy = -(x+1)
(2-x)y = -(x+1)
y = (x+1)/(x-2)
Now try a check
say x = 3 (do not use x = 1, undefined)
then
y = 7/2
now see if that reverses
if x = 7/2
y = (7/2 +1)/ (7/2 -2)
= (9/2) / (3/2)
y = 3
sure enough
that's a great anecdote, but a true verification means that you have to show that
(2[(x+1)/(x-2)]+1)/([(x+1)/(x-2)]-1) = x
Thank you guys the help is great
To find the inverse of a function, we need to switch the roles of x and y and solve for y. In other words, we need to solve the equation x = 2y + 1 / y - 1 for y, and then we can express y as a function of x.
1. Start by switching x and y in the equation:
x = 2y + 1 / y - 1
2. Multiply both sides of the equation by y - 1 to eliminate the denominator:
x(y - 1) = 2y + 1
3. Distribute the x on the left side:
xy - x = 2y + 1
4. Move all the terms involving y to one side and all the constant terms to the other side:
xy - 2y = x + 1
5. Factor out y on the left side:
y(x - 2) = x + 1
6. Divide both sides of the equation by (x - 2) to solve for y:
y = (x + 1) / (x - 2)
Thus, the inverse of the function f(x) = (2x + 1) / (x - 1) is given by g(x) = (x + 1) / (x - 2).
To verify if g(x) is indeed the inverse of f(x), we can check whether f(g(x)) = x and g(f(x)) = x.
1. f(g(x)):
Start with g(x) = (x + 1) / (x - 2)
Substitute g(x) into f(x):
f(g(x)) = 2 * [(x + 1) / (x - 2)] + 1 / [(x + 1) / (x - 2)] - 1
Simplify the expression further.
2. g(f(x)):
Start with f(x) = (2x + 1) / (x - 1)
Substitute f(x) into g(x):
g(f(x)) = [(2x + 1) / (x - 1) + 1] / [(2x + 1) / (x - 1) - 2]
Simplify the expression further.
Calculate both f(g(x)) and g(f(x)) to check if they simplify to x. If both expressions simplify to x, then g(x) is indeed the inverse of f(x).