find f^-1 for the function f(x)= 3sqr2x-1
A. f^-1(x)=1/2(x^3+1)
B. f^-1(x)=1/2(x+1)^3
C.f^-1(x)=(x/2+1)^3
D. f^-1(x)=(x/2)^3+1
To find the inverse of a function, we can follow the steps below:
1. Start with the function f(x) = 3√(2x-1).
2. Replace f(x) with y: y = 3√(2x-1).
3. Swap x and y: x = 3√(2y-1).
4. Solve for y:
Divide both sides of the equation by 3: x/3 = √(2y-1).
Square both sides of the equation: (x/3)^2 = 2y-1.
Add 1 to both sides of the equation: (x/3)^2 + 1 = 2y.
Divide both sides of the equation by 2: (x/3)^2/2 + 1/2 = y.
5. Replace y with f^-1(x): f^-1(x) = (x/3)^2/2 + 1/2.
Therefore, the answer is D. f^-1(x) = (x/3)^2/2 + 1/2.
To find the inverse of the function f(x) = 3√(2x - 1), we can follow these steps:
Step 1: Replace f(x) with y.
y = 3√(2x - 1)
Step 2: Swap x and y.
x = 3√(2y - 1)
Step 3: Solve for y.
x = 3√(2y - 1)
Cube both sides:
x^3 = 27(2y - 1)
x^3 = 54y - 27
54y = x^3 + 27
y = (x^3 + 27)/54
Therefore, the inverse of the function f(x) = 3√(2x - 1) is:
f^-1(x) = (x^3 + 27)/54
So, the correct answer is A. f^-1(x) = 1/2(x^3 + 1).
show the steps
Apologies for the confusion in my previous response.
To find the inverse of the function f(x) = 3√(2x - 1), we can follow these steps:
Step 1: Replace f(x) with y.
y = 3√(2x - 1)
Step 2: Swap x and y.
x = 3√(2y - 1)
Step 3: Solve for y.
x = 3√(2y - 1)
Cube both sides to eliminate the square root:
x^3 = (3√(2y - 1))^3
x^3 = 27(2y - 1)
Simplify the right side:
x^3 = 54y - 27
Isolate y by moving the constant term to the other side:
54y = x^3 + 27
Divide both sides by 54 to solve for y:
y = (x^3 + 27)/54
Therefore, the inverse of the function f(x) = 3√(2x - 1) is:
f^-1(x) = (x^3 + 27)/54
So, the correct answer is A. f^-1(x) = 1/2(x^3 + 1).
To find the inverse of a function, f(x), you need to swap the variables x and y and solve for y. Let's apply this process to the given function, f(x) = 3√(2x-1).
Step 1: Swap the variables:
x = 3√(2y-1)
Step 2: Isolate the variable:
x/3 = √(2y-1)
Step 3: Square both sides to remove the square root:
(x/3)^2 = 2y-1
Step 4: Solve for y:
2y-1 = (x/3)^2
2y = (x/3)^2 + 1
y = (1/2)(x/3)^2 + 1/2
Now we have y in terms of x, which is the inverse of f(x). So, the answer is:
f^(-1)(x) = (1/2)(x/3)^2 + 1/2
Looking at the given options:
A. f^(-1)(x) = 1/2(x^3 + 1)
B. f^(-1)(x) = 1/2(x + 1)^3
C. f^(-1)(x) = (x/2 + 1)^3
D. f^(-1)(x) = (x/2)^3 + 1
The correct answer is NOT listed among the options.