Question

Graph the following absolute value functions and write the corresponding piecewise functions for each.
g(x)=|x^2-4|+1
Piecewise:

Answer 1

Can someone please help me graph this and find the piecewise function

Answer 2

this is going to be confusing

Answer 3

yes

Answer 4

the absolute value of \(\spadesuit\) is \(\spadesuit\) if \(\spadesuit>0\) otherwise it is \(-\spadesuit\)

Answer 5

of what ?

Answer 6

that is a spade

Answer 7

ok

Answer 8

lets write is as a piecewise function

Answer 9

alright

Answer 10

\[f(x) = |x| = \left\{\begin{array}{rcc}
-x & \text{if} & x<0 \\
x& \text{if} & x\ge0
\end{array}
\right.\]

Answer 11

okay

Answer 12

if x is negative, then the absolute value of \(x\) is \(-x\) (which is positive) and if \(x\) is positive then the absolute value of \(x\) is just \(x\)

Answer 13

no you don't have \(x\) you have \(x^2-4\) right?

Answer 14

oh okay

Answer 15

yes

Answer 16

so lets replace \(x\) in the above definition \[f(x) = |x| = \left\{\begin{array}{rcc}
-x & \text{if} & x<0 \\
x& \text{if} & x\ge0
\end{array}
\right.\] by \(x^2-4\)

Answer 17

it is easy for me to do, i will just cut and paste

Answer 18

okay

Answer 19

\[f(x) = |x^2-4| = \left\{\begin{array}{rcc}
-(x^2-4) & \text{if} & x^2-4<0 \\
x^2-4& \text{if} & x^2-4\ge0
\end{array}
\right.\]

Answer 20

we are clearly not done yet, but almost

Answer 21

alright

Answer 22

we just have to write the two inequalities above in terms of \(x\) instead of \(x^2-4\)
first is it clear what i did? i literally copied and pasted \(x^2-4\) in to each \(x\)

Answer 23

Yeah I see what you did

Answer 24

ok good
now how to we solve \[x^2-4\geq 0\]? do you know?

Answer 25

Do we plug in numbers for x?

Answer 26

no

Answer 27

\(y=x^2-4\) is a parabola that opens up
it is zero if \(x=-2\) or if \(x=2\)

Answer 28

why -2 or 2 ?