find f^-1 for the function f(x)= 3sqr 2x-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 function, we need to solve the equation y = 3√(2x - 1) for x.
1. Start by swapping the x and y variables: x = 3√(2y - 1).
2. Cube both sides of the equation to eliminate the cube root: x^3 = (3√(2y - 1))^3.
3. Simplify the right side: x^3 = 27(2y - 1).
4. Divide both sides by 27: (x^3)/27 = 2y - 1.
5. Add 1 to both sides: (x^3)/27 + 1 = 2y.
6. Divide both sides by 2: (x^3)/54 + 1/2 = y.
Therefore, the inverse function of f(x) = 3√(2x - 1) is f^(-1)(x) = (x^3)/54 + 1/2.
None of the provided answer choices match the correct inverse function.
To find the inverse of the function f(x) = 3√(2x - 1), we need to determine the value of x when given f(x) as input.
Step 1: Start with the equation f(x) = 3√(2x - 1).
Step 2: Replace f(x) with y to obtain y = 3√(2x - 1).
Step 3: Swap x and y to get x = 3√(2y - 1).
Step 4: Solve the equation for y.
- Cube both sides of the equation to eliminate the cube root: (x/3)^3 = 2y - 1.
- Simplify: x^3/27 = 2y - 1.
- Add 1 to both sides of the equation: x^3/27 + 1 = 2y.
- Divide both sides of the equation by 2: (x^3/27 + 1)/2 = y.
- Simplify further: (x^3/27 + 1)/2 = y.
Step 5: Replace y with f^(-1)(x) to obtain the inverse function f^(-1)(x) = (x^3/27 + 1)/2.
Therefore, the correct answer is option C: f^(-1)(x) = (x/2 + 1)^3.
To find the inverse of a function, we need to swap the roles of the input variable and the output variable. In other words, we need to find an expression for x in terms of f(x).
Let's start with the given function:
f(x) = 3√(2x - 1)
To find the inverse, we want to solve for x. Let's replace f(x) with y:
y = 3√(2x - 1)
Now, let's switch the roles of x and y:
x = 3√(2y - 1)
We can now solve this equation for y. Start by isolating the radical term:
x = 3√(2y - 1)
(x/3)^2 = 2y - 1
(x^2/9) + 1 = 2y
(x^2/9) = 2y - 1
2y - 1 = (x^2/9)
Now, let's solve for y:
2y = (x^2/9) + 1
y = (x^2/18) + 1/2
Now that we have y in terms of x, we can replace y with f^-1(x):
f^-1(x) = (x^2/18) + 1/2
Therefore, the correct answer is option C. f^-1(x) = (x/2 + 1)^3.