Let f(x) =(x+3)/(2x-5)

Find the inverse function of f

What is the domain of f? What is the domain of the inverse function?

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To find the inverse function of f(x), we need to switch the roles of x and f(x) and solve for x.

1. Start with the equation f(x) = (x + 3)/(2x - 5).
2. Replace f(x) with y: y = (x + 3)/(2x - 5).
3. Swap x and y: x = (y + 3)/(2y - 5).
4. Solve for y: Multiply both sides by 2y - 5: x(2y - 5) = y + 3.
5. Distribute x: 2xy - 5x = y + 3.
6. Rearrange the equation: 2xy - y = 5x + 3.
7. Factor out y: y(2x - 1) = 5x + 3.
8. Divide both sides by (2x - 1): y = (5x + 3)/(2x - 1).

The inverse function of f(x) is given by f^(-1)(x) = (5x + 3)/(2x - 1).

Now, let's determine the domain of f(x):

The domain of a function refers to all the possible values of x for which the function is defined. In this case, the function f(x) = (x + 3)/(2x - 5) will be defined as long as the denominator 2x - 5 is not equal to zero (to avoid division by zero). Therefore, the domain of f(x) is all real numbers except for x = 5/2.

So, the domain of f(x) is (-∞, 5/2) U (5/2, +∞).

Now, let's determine the domain of the inverse function:

To find the domain of the inverse function, we need to consider the original range of the function f(x). Since the original function is defined for all real numbers except x = 5/2, the range of the original function will be all real numbers except f(5/2).

Calculating f(5/2), we get:
f(5/2) = (5/2 + 3)/(2(5/2) - 5) = 13/4

Therefore, the inverse function will have a domain of all real numbers except for x = 13/4.

So, the domain of the inverse function f^(-1)(x) is (-∞, 13/4) U (13/4, +∞).

Let f(x) = y

Then solve for x in terms of y.
(2x-5)y = x + 3
2xy -3 = 5y + x
x(2y -1)= 5y + 3
x = (5y+3)/(2y-1)
That is the inverse function. It can be written
f^-1(x)= (2x+3)/(2x-1)
The domain of f(x) is all numbers except 5/2.
The domain of f^-1(x) is all numbers except 1/2

Thank you for showing me! Now I can do the rest on my own hopefully without a problem! Thanks again!