about finding the inverse of y=(2x+1)/(x+3).
should you multiply (x+3) on both sides and get y(x+3)=2x+1?
then what would be your next step?
This one is a little tricky because we we have a rational function.
What we want to end up with is x(y)
To do this you can interchange x and y and try to solve for x. That function is the inverse. Thus:
y=(2x+1)/(x+3)
x=(2y+1)/(y+3), then solve for y
(y+3)x= (2y+1)
yx+3x = 2y+1
3x-1=2y-yx =
3x-1= y(2-x) =
y= (3x-1)/(2-x)
Now interchange the x and y again. (after isolating y)
y = (3x-1)/(2-x)
And if we did things correctly, we should have the inverse function. Test it by composing functions.
(3((2x+1)/(x+3) -1))/(2-(2x+1)/(x+3))=
The numerator is:((6x+3)-(x+3))/(x+3))=
5x/(x+3)
The denominator is:(2(x+3)-(2x+1))/(x+3)=
5/(x+3)
Putting them together we have 5x/(x+3) times (x+3)/5 = x.
Thus they are inverses. Test it by graphing too.
That makes sense! Thank you so much!!!
comment:I don't think the second interchange is necessary if we want y(x). It's been awhile since I worked this, so check my work and ask questions if anything isn't clear.
You're correct! Apologies for the confusion. In this case, if we want to find the inverse function y(x) of y = (2x + 1) / (x + 3), we can follow a different approach without the need for a second interchange of x and y.
To find the inverse function, we start with the original equation:
y = (2x + 1) / (x + 3)
Step 1: Swap x and y:
x = (2y + 1) / (y + 3)
Step 2: Solve for y:
Multiply both sides of the equation by (y + 3) to eliminate the denominator:
x(y + 3) = 2y + 1
Expand and rearrange terms:
xy + 3x = 2y + 1
Step 3: Isolate the y term:
Subtract xy from both sides:
3x = 2y - xy + 1
Rearrange terms:
3x - 1 = y(2 - x)
Step 4: Solve for y:
Divide both sides of the equation by (2 - x):
y = (3x - 1) / (2 - x)
And there you have it! The inverse function of y = (2x + 1) / (x + 3) is y = (3x - 1) / (2 - x).
Remember to always test the inverse function by composing it with the original function. In this case, substituting y = (3x - 1) / (2 - x) into y = (2x + 1) / (x + 3) should result in x.