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Consider the following. f(x)=(8x^(3)-x^(2)+7)/(3x^(3)+24) (a) Find the domain of the function

User Sherman
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In order to find the domain of the given function, we need find for which x-values the function is defined. The issue would be if the denominator equals 0, which would make the function undefined.

So, we have to find the solutions of equation 3x^3 + 24 = 0.

To solve for x, we need to get 3x³ on its own. We do this by subtracting 24 from both sides of the equation to isolate 3x^3:

3x^3 = -24

Then we divide both sides of the equation by 3 to isolate x^3:

x^3 = -24/3

We simplify the right-hand side to get:

x^3 = -8

Now, to solve for x, we could take the cube root of both sides to get

x = -2

So the function is not defined when x = -2.

Now let's figure out the domain. The domain of the function will be all real numbers except for x = -2. In interval notation, we can represent this as (-∞, -2) U (-2, ∞). "U" denotes the union of two sets, meaning that the domain includes all numbers from negative infinity to -2, and from -2 to positive infinity, but does not include -2 itself.

So, the domain of the function is all real x, except x=-2.

User Makoto Miyazaki
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