Question

write the absolute value function as a piecewise function. y=I4x+1I+2x-3

Answer 1

We need the 'turning' point of your absolute value function, that is, the value of x which would make the thing inside the bars || equal to zero.

That said, when is 4x + 1 equal to zero?

Answer 2

@terenzreignz when x=-1/4 ?

Answer 3

Precisely :)
So that's going to be the turning point of your function :D
\[\Large f(x) = \left\{\begin{matrix}&\qquad x \le -\frac14\\\\\\\\\\&\qquad x> -\frac14\end{matrix}\right.\]

Answer 4

Now, when x is less than -1/4
4x+1 is positive or negative?

Answer 5

@terenzreignz negative?

Answer 6

Yes.
So, since it's negative, and in absolute value, it becomes positive when x is less than -1/4
Then we multiply it by -1 and it becomes
-4x - 1
okay?

Answer 7

ok that makes sense so far

Answer 8

Right?
Because |4x + 1| is basically the same as -4x - 1 whenever x is less than -1/4

Answer 9

lol okay, so when x is less than or equal to -1/4, then the thing inside the absolute value gets multiplied by -1 (to turn it positive)
\[\Large f(x) = \left\{\begin{matrix}\color{blue}{-4x-1}+2x -3&\qquad x \le -\frac14\\\\\\\\\\&\qquad x> -\frac14\end{matrix}\right.\]
We will simplify later.
Anyway, when x is greater than -1/4
then |4x + 1| is positive, yes?

Answer 10

Rather, sorry (since |4x+1| is always positive anyway)
when x is greater than -1/4
then 4x + 1 is positive, right?

Answer 11

oh ok!!! so f(x)={ 6x-2 when x greater than or equal to -1/4
-2x-4 when x < -1/4

Answer 12

@terenzreignz

Answer 13

Someone's eager :)
LOL
Correct :)
\[\Large f(x) = \left\{\begin{matrix}-2x-4&\qquad x \le -\frac14\\\\\\\\\\6x-2&\qquad x> -\frac14\end{matrix}\right.\]
Note that it doesn't matter if we switch the "or equal to" like this
\[\Large f(x) = \left\{\begin{matrix}-2x-4&\qquad x \color{red}< -\frac14\\\\\\\\\\6x-2&\qquad x\color{red}{\ge}-\frac14\end{matrix}\right.\]
Since both 'pieces' of the function have the same value when x = -1/4

Great job, by the way ^_^

Answer 14

ok thank you so much! i finally understand this stuff now! :)

Answer 15

Glad to hear it :)
Signing off for now...
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Terence out