Factoring Question

[mstudent]

Hello

I am encountering a problem in an assignment for my pre-calculus course.
It is as follows:

I am to find the function of a function, in this case, g of g, where g is x^2 - 4x

I have worked out the problem to the point where I arrive at x^4 - 8x^3 +12x^2 + 16x, which I am confident is correct.

Now I am to factor and find the domain of this function; I would appreciate a hint as to how I should begin factoring.

Also, to the best of my knowledge, domains of functions can be described as the denominators values for which the function is not solvable. Or, perhaps this is better described as values for the unknown variable (in this case 'x') for which the equation is unsolvable. Is this right?

Thank-you again.

Greg Z

 

[soroban]

Hello, mstudent!

Your mathematical grammar is sloppy . . .

I am encountering a problem in an assignment for my pre-calculus course.
It is as follows:

I am to find the function of a function.
In this case, \(\displaystyle g\circ g\), where \(\displaystyle g(x) \,=\,x^2 - 4x\)

I have worked out the problem to: .\(\displaystyle x^4 - 8x^3 +12x^2 + 16x\) .Correct!

Now I am to factor and find the domain of this function. .[1]
I would appreciate a hint as to how I should begin factoring.

Also, to the best of my knowledge, domains of functions can be described as
. . the denominators values for which the function is not solvable. .[2]
Or, perhaps this is better described as values of the variable for which the equation is unsolvable. .[3]
Is this right?

[1] . Why do you have to factor?


[2] . Puzzling phrases . . .

. . . What are "denominator values"?

. . . "The function is not solvable"
. . . We don't solve functions.


[3] . "The equation is unsolvable" . . . What equation?
. . . . \(\displaystyle \text{Solving }f(x) = 0\,\text{ gives us the }x\text{-intercepts . . . So what?}\)


I believe I know what you mean.

The domain of \(\displaystyle f(x)\) consists of all real numbers except:
. . values of \(\displaystyle x\) for which \(\displaystyle f(x)\) does not exist or is undefined.


Example: .\(\displaystyle f(x) \:=\:\sqrt{x-3}\)

If \(\displaystyle x < 3\), the function is a complex (imaginary) number.

. . The domain is: .\(\displaystyle x \ge 3\:\text{ or }\:[3,\infty)\)


Example: .\(\displaystyle f(x) \:=\:\dfrac{1}{x-2}\)

If \(\displaystyle x = 2\), the function is undefined.

. . The domain is: .\(\displaystyle \{\text{all real numbers}\ne 2\}\:\text{ or }\: (\text{-}\infty, 2) \cup (2,\infty)\)