I need to know if I have done this correctly.
Problem
If f is one-to-one, find an equation for its inverse.
f(x)=x^3-5
y=x^3-5
x=y^3-5
5+x=y^3-5+5
5+x=y^3
My answer:
y^3=5+x
3�ãy=3�ã5+x
y=3�ã5+x
f^-1(x)3�ã5+x
y = (5+x)^(1/3)
is the easy way to write that.
Now lets say x = 3
then
y = 8^1/3 = 2
but what if y = (-1 + i sqrt 3)
square that
1 -2 i sqrt 3 - 3 = (-2 - 2i sqrt 3)
multiply that by (-1+ i sqrt 3)
2 -2 i sqrt 3 + 2 i sqrt 3 + 6
= 8 !!!!!
I also claim that the complex conjugate
y = (-1 -i sqrt 3)
cubed is also 8
therefore I claim that y = 2 or y = -1+i sqrt 3 or y = -1 -i sqrt 3
are all values of y for x = 3
So if your algebra subject is about complex as well as real numbers, there are three values of y for any value of x in the inverse problem.
Sorry, my answer is:
y^3=5+x
3�ãy=3�ã5+x
y=3�ã5+x
f^-1(x)=3�ã5+x
but whatever your symbol is for the cube root, you have to take the cube root of the whole right side, need parentheses
(5+x)
To determine if you have found the correct inverse function, we can substitute the values of f(x) and f^(-1)(x) into both equations and see if they simplify to the same form.
Starting with the original function f(x) = x^3 - 5:
Step 1: Replace f(x) with y
y = x^3 - 5
Step 2: Swap x and y
x = y^3 - 5
Step 3: Solve for y
x + 5 = y^3
Taking the cube root of both sides:
y = (x + 5)^(1/3)
Now let's substitute the values into the equation you obtained for the inverse function:
f^(-1)(x) = 3√(5 + x)
By comparing the two expressions, we can see that they are equivalent. So yes, you have correctly found the equation for the inverse function of f(x).