given: f(x)=(2x+3)/(3X+2),
find the inverse function of this.
please don't give me the answer only, i would like to see the full working out so then i'll know how to do it next time!thanks!!:-)
What you want to do is treat this function as y=(2x+3)/(3x+2)
Now replace every y by x, and every x by y.
Next, alter the equation in that way, so that it read x= some function of y.
When you have done all that, just replace the y's by x, and you have your inverse function in terms of x
I hate to say this, but I found your explanation very difficult to understand, especially your final step.
The method is quite simple :
Step 1. form the inverse equation by interchanging the x and y variables, just like you said.
Step 2. Solve this new equation for y.
That's all, you are done.
To find the inverse of a function, we need to solve for the input variable in terms of the output variable. In this case, we will solve for x in terms of y.
Given the function f(x) = (2x + 3)/(3x + 2), we want to find an expression for x in terms of y. Let's start by replacing f(x) with y to make the problem more clear:
y = (2x + 3)/(3x + 2)
Step 1: Swap x and y
x = (2y + 3)/(3y + 2)
Step 2: Solve for y
To solve for y, we need to isolate it on one side of the equation. We will begin by multiplying both sides of the equation by (3y + 2):
x(3y + 2) = 2y + 3
Step 3: Expand and simplify
3xy + 2x = 2y + 3
Step 4: Move variables to one side
3xy - 2y = 3 - 2x
Step 5: Factor out y
y(3x - 2) = 3 - 2x
Step 6: Solve for y
y = (3 - 2x)/(3x - 2)
Now we have an expression for y in terms of x. Therefore, the inverse function of f(x) = (2x + 3)/(3x + 2) is given by f^(-1)(x) = (3 - 2x)/(3x - 2).
Note: It is always a good idea to check if the inverse function is valid by verifying that when you compose both functions, you get the identity function. In this case, you can confirm this by computing (f^-1 ∘ f)(x) and (f ∘ f^-1)(x) and checking if they both simplify to x.