Find f^-1 in similar form for each of the following functions.
Question no. 1 ↓
f:x→3x over x-3,x not equal to 3
Question no. 2 ↓
f:x→2x+4 over 2x-1,x not equal to 1over2
I will do the 2nd, you do the 1st in the same way
f(x) = (2x+4)/(2x-1)
or
y = (2x+4)/(2x-1)
inverse by switching x's and y's
x = (2y+4)/(2y-1)
simplify and solve this for y ....
2xy - x = 2y+4
2xy - 2y = x+4
y(2x - 2) = x+4
y = (x+4)/(2x-2)
f^-1 (x) = (x+4)/(2x-2) , x ≠ 1
I usually test my answer, let x = 3 , (I get no fractions)
f(3) = 10/5 = 2
f^-1 (2) = 6/2 = 3
(this does not "prove" that my answer is right, but if I did not get 3 as my return answer, by equation would have been wrong. My test "strongly suggests" that I am right)
Reiny,thanks a lot! You are right...can you explain why the answer x is not equal to 1 ?
What would happen if you sub x = 1 into our new denominator?
Can we divide by zero ?
Ohh...ok..thank you!
No.1
y=3x/x-3
x=3y/y-3
xy-3y=3x
y(x-3)=3x
y=3x/x-3
f^-1(x)=3x/x-3,x≠3
Am i correct ?:D
yes, but in this form of typing you have to use brackets.
f^-1 (x) = 3x/(x-3) or else the 3x is divided by x only
To find the inverse function, f^(-1), for each of these functions, we need to follow these steps:
Step 1: Write the function in terms of y instead of x.
Step 2: Replace y with x.
Step 3: Solve for x.
Step 4: Replace x with y and state the inverse function.
Now let's find f^(-1) for each function separately:
Question 1:
The given function is f(x) = 3x / (x - 3), where x is not equal to 3.
Step 1: Write the function in terms of y.
y = 3x / (x - 3)
Step 2: Replace y with x.
x = 3y / (y - 3)
Step 3: Solve for x.
x(y - 3) = 3y
xy - 3x = 3y
xy - 3y = 3x
xy - 3y - 3x = 0
y(x - 3) - 3x = 0
y(x - 3) = 3x
y = 3x / (x - 3)
Step 4: Replace x with y and state the inverse function.
f^(-1)(x) = 3x / (x - 3), where x is not equal to 3.
Question 2:
The given function is f(x) = (2x + 4) / (2x - 1), where x is not equal to 1/2.
Step 1: Write the function in terms of y.
y = (2x + 4) / (2x - 1)
Step 2: Replace y with x.
x = (2y + 4) / (2y - 1)
Step 3: Solve for x.
x(2y - 1) = 2y + 4
2xy - x = 2y + 4
2xy - 2y = x + 4
2y(x - 1) = x + 4
y(x - 1) = (x + 4) / 2
y = (x + 4) / (2(x - 1))
Step 4: Replace x with y and state the inverse function.
f^(-1)(x) = (x + 4) / (2(x - 1)), where x is not equal to 1/2.
So, the inverse functions for the given functions are:
Question 1:
f^(-1)(x) = 3x / (x - 3), where x is not equal to 3.
Question 2:
f^(-1)(x) = (x + 4) / (2(x - 1)), where x is not equal to 1/2.