# Find the inverse of y=x^3 + x +4

## Homework Statement

I'm helping my friend review for his exam in precalc and I can't seem to get this inverse.

Find the inverse of y=x^3 + x +4

## The Attempt at a Solution

I switched the variables and then tried to isolate the y but I failed.
I ended with x-4=y^3+y I don't know how to isolate the y from here how would you go about it?

symbolipoint
Homework Helper
Gold Member
you would obtain x=y^3 +y +4, but unless you can find a way to produce an equivalent cubed binomial for the right side, you are stuck. If there are more than one y value for any single x value, then that is not an inverse function.

Given f (x) = x3 + x + 4 , find f -1 (1) accurate to 2 decimal places. Is the exact question I thought I would make it easier by just asking to find the inverse on here...

symbolipoint
Homework Helper
Gold Member
If you are asking for how to evaluate the inverse of f(x) at x=1, then you want to start with the inverse relationship, x= y^3 + y + 4 [I assume you still meant that the original function was a cubic], and substitute 1=x; and then find the possible values for y. If you obtain two values of y, then the inverse of f(x) is not a function.

1 = y^3 + y + 4
y^3 + y + 3 = 0
Complete the square or use quadratic formula solution;
What is y?

symbolipoint
Homework Helper
Gold Member
What I said in here (below) is ridiculous. You can't complete the square for a cubic; although you can look for the presence of any x which gives more than one y value.

If you are asking for how to evaluate the inverse of f(x) at x=1, then you want to start with the inverse relationship, x= y^3 + y + 4 [I assume you still meant that the original function was a cubic], and substitute 1=x; and then find the possible values for y. If you obtain two values of y, then the inverse of f(x) is not a function.

1 = y^3 + y + 4
y^3 + y + 3 = 0
Complete the square or use quadratic formula solution;
What is y?

HallsofIvy
Homework Helper

## Homework Statement

I'm helping my friend review for his exam in precalc and I can't seem to get this inverse.

Find the inverse of y=x^3 + x +4

## The Attempt at a Solution

I switched the variables and then tried to isolate the y but I failed.
I ended with x-4=y^3+y I don't know how to isolate the y from here how would you go about it?

Given f (x) = x3 + x + 4 , find f -1 (1) accurate to 2 decimal places. Is the exact question I thought I would make it easier by just asking to find the inverse on here...

That's a completely different question! In order to solve the equation y3+ y+ 4= x, for any x, you would have to use Cardano's cubic formula- and that's extremely complex!

What you are asked to do here is to use some numerical method to solve the equation y3+ y+ 4= 1 or y3+ y+ 3= 0. A simple way to solve such an equation is "bijection". If y= -2, (-2)3+ (-2)+ 3= -8- 2+ 3= -7< 0. If y= -1, (-1)3+ (-1)+ 3= 1> 0 Since the value at x= -2 is negative and the value at x= -1 is positive, there must be a solution between -2 and -1. Try half way between: -3/2. Is (-3/2)3+ (-3/2)+ 3 positive or negative? Whichever it is, there must be a root between -3/2 and either -2 or -1 depending on which gave the opposite sign. Try half-way between just because it is convenient.