The following function is one-to-one; find its inverse
f(x)=x^3+5
I know that the first step would be to set y = to x so I did y=x^3+5
Then you swap x with y so x=y^3+5.
Then you solve for y and this is where I think i messed up. I did
y^3=x-5 = y=x^3-5
Then the inverse would be f^-1(x)=x^3-5
Is this correct or did I do that last step wrong?
Your error is in solving for y in
x = y^3 + 5
y^3 = x-5 , you had that
y = cuberoot(x-5) or (x-5)^(1/3)
check:
let x = 2 in the original y = x^3+5
y = 2^3+5 = 13
now let x = 13 in the invers
y = cuberoot(13-5
= cuberoot(8)
= 2
which is what we started with
So what is the inverse?
I stated it ....
y = cuberoot(x-5)
or
y = (x-5)^(1/3)
or
f(x) = (x-5)^(1/3)
O ok I understand. Thanks!!
Your understanding of the steps to find the inverse function is correct, but you made a mistake in the last step. Let's go through the process again and correct the mistake.
To find the inverse of the function f(x) = x^3 + 5, follow these steps:
1. Start with the equation: y = x^3 + 5.
2. Swap the variables x and y: x = y^3 + 5.
3. Solve the equation for y. Begin by subtracting 5 from both sides: x - 5 = y^3.
4. Take the cube root of both sides to isolate y: ∛(x - 5) = y.
5. Finally, replace y with f^(-1)(x) to represent the inverse function: f^(-1)(x) = ∛(x - 5).
So, the correct inverse function is f^(-1)(x) = ∛(x - 5).