can you help me find the inverse of this function algebraically please?

f(x) = 2x^3 + 1

f-1 = 2x^3

(f-1)/2 = x^3
x = ∛((f-1)/2)

so,

f^-1(x) = ∛((x-1)/2)

to find the inverse of a function, rewrite it in the form

y = .....
interchange the x and y variable,
then solve this new equation for y

so
y = 2x^3 + 1
inverse: x = 2y^3 + 1

2y^3 = x - 1
y^3 = (x - 1)/2

y = ( (x-1)/2 )^(1/3)

check with a given point,
in original let x = 3 , y = 2(27) + 1 = 55
in inverse let, x = 55 , y = ( (55-1)/2 )^(1/3)
= 27^(1/3) = 3
My answer is "more than likely" correct

Sure! To find the inverse of a function algebraically, we need to switch the roles of x and y and solve for y. Let's start by replacing f(x) with y:

y = 2x^3 + 1

Now, we'll swap x and y:

x = 2y^3 + 1

Next, we'll solve this equation for y. Let's follow these steps:

Step 1: Subtract 1 from both sides:

x - 1 = 2y^3

Step 2: Divide both sides by 2:

(x - 1)/2 = y^3

Step 3: Take the cube root of both sides:

∛((x - 1)/2) = y

So, the inverse function is:

f^(-1)(x) = ∛((x - 1)/2)

Remember, the cube root symbol (∛) represents raising a number to the power of 1/3.