# Find the domain of this function

• calif2a8
In summary, the domain of sqrt(2x-x^3) is (-inf,-sqrt(2)] and [0, sqrt(2)]. To find this domain, you need to factor the quadratic part and draw the graph of y=x^3-2x and y=x^2-2 to determine where the function is positive or negative. Dividing by x in the initial equation will change the graph and lead to incorrect solutions.
calif2a8
What is the domain of sqrt(2x-x^3)

I thought this would be pretty straight forward but I am completely stumped. So obviously square root is not defined for (-inf, 0), therefore I said:

2x-x^3 >= 0

Working this out you get x <= sqrt(2) as the domain.

That is partially correct but I can't for the life of me get the rest of the answer.
The final answer is (-inf,-sqrt(2)] and [0, sqrt(2)]

HOW do you find out that (-sqrt(2), 0) is not defined?

Thanks for the read and help :D

calif2a8 said:
What is the domain of sqrt(2x-x^3)

I thought this would be pretty straight forward but I am completely stumped. So obviously square root is not defined for (-inf, 0), therefore I said:

2x-x^3 >= 0

Working this out you get x <= sqrt(2) as the domain.
You oversimplified things. I think this is what you did:
2x - x3 ≥ 0

x(2 - x2) ≥ 0 (so far, so good)

Next, it looks like you divided both sides by x - not good.

You need to factor the quadratic part, and you can't throw away the x factor as you did. Note that x = -2 is in the domain, which your work doesn't show.
calif2a8 said:
That is partially correct but I can't for the life of me get the rest of the answer.
The final answer is (-inf,-sqrt(2)] and [0, sqrt(2)]

HOW do you find out that (-sqrt(2), 0) is not defined?

Thanks for the read and help :D

Mark44 said:
You oversimplified things. I think this is what you did:

x(2 - x2) ≥ 0 (so far, so good)

Actually I did something worse i think, but I didn't know it was not correct? I started with

2x - x3 ≥ 0

but then I did this: 2x ≥ x3 and then canceled the Xs...
so then I was left with

2 ≥ x2 ... leading to my original answer of sqrt(2) ≥ x

Why is this approach wrong?

Thanks for all the replies by the way :)

calif2a8 said:
Actually I did something worse i think, but I didn't know it was not correct? I started with

2x - x3 ≥ 0

but then I did this: 2x ≥ x3 and then canceled the Xs...
so then I was left with

2 ≥ x2 ... leading to my original answer of sqrt(2) ≥ x

Why is this approach wrong?

Thanks for all the replies by the way :)

Dividing through by x before or after moving the x3 to the other side is an equivalent procedure.

What you should do is draw the graph of $y=x^3-2x$ and notice where y>0. You should know that $y=x(x^2-2)$ is the same graph. Now draw just $y=x^2-2$. Although the shape is completely different, the part where it crosses the x-axis at $x=\pm\sqrt{2}$ is still the same with both graphs, but it's missing some extra information, such as it doesn't cross the x-axis at x=0.

Why's this? Well, when you divided $x^3-2x$ by x, this assumes that $x\neq 0$ because you can't divide by zero, but x can be equal to zero, so you've changed the graph.

By the way, if $x^2\leq 2$ then the solution set for this inequality is [-2,2]. Again, draw the graph of $y=x^2-2$ and find where y<0.

## 1. What is the definition of a domain in a function?

The domain of a function is the set of all possible input values for the function. It represents the range of values that the independent variable can take on.

## 2. How do you determine the domain of a function?

To determine the domain of a function, you need to look at the given function and identify any restrictions or rules for the input values. These restrictions can include non-permissible values such as negative numbers under a square root or division by zero.

## 3. Can the domain of a function be negative?

Yes, the domain of a function can include negative numbers as long as there are no restrictions or rules that prevent it. For example, the domain of the function f(x) = x² is all real numbers, including negative numbers, but the function g(x) = 1/x has a restricted domain of all real numbers except for x = 0.

## 4. What happens if there is no specified domain for a function?

If there is no specified domain for a function, it is assumed to have a domain of all real numbers. However, it is important to note any potential restrictions or rules that may apply to the function.

## 5. Can the domain of a function change?

Yes, the domain of a function can change depending on the given context or situation. For example, a rational function may have a different domain when it is graphed on a certain interval compared to its overall domain. It is important to consider the context when determining the domain of a function.

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