I have some question about absolute value (how convert it in another form correctly)?

Question

Hi maths people I write next week a test in linear algebra and I have question because when asked in test I want do it good.

We have for example $|x^2-1| + |x^2-1|$.

How can I write it in another way? Is this correct? $|2(x^2-1)|$ or is this correct $2(|x^2-1|)$. I think first is correct because from beginning we have all inside absolute value sign and so it must all be in absolute value sign after. But no sure..

What when we have multiplication instead of addition? $$|x^2-1| \cdot |x^2-1| = |(x^2-1)^2| \text{ ?}$$

Please need help I have no idea what is good and what not..

Answer 1

it is $$|x^2-1|+|x^2-1|=2|x^2-1|$$ like $$a+a=2a$$

Answer 2

The expressions $$ |2(x^2-1)| $$ and $$ 2(|x^2-1|) $$ are completely equivalent and it's just a matter of taste which one to use; the latter is usually written $2|x^2-1|$, because the bars of the absolute value work as brackets.

What about $|x^2-1|\cdot|x^2-1|$? Well, both $$ |x^2-1|^2 $$ and $$ (x^2-1)^2 $$ are good, because of $(-a)^2=a^2$. Note that the even exponent is important: you're not allowed to write $$ |x^2-1|\cdot|x^2-1|\cdot|x^2-1|= (x^2-1)^3 $$ because in general $(-a)^3\ne a^3$.

Answer 3

Hint.-See about the function $f(x)=\sqrt{x^2}$. You know that $$f(x)+f(x)=2\sqrt{x^2}\space \space \text{ and }\space \space f(x)\cdot f(x)=x^2$$

Compare now $f(x)$ with $g(x)$. Are both functions the same? Look at the graph of both functions.