Thread: how to draw this function?

1. how to draw this function?

$\displaystyle \frac{3-3 ^{x}}{ \left|3-3 ^{x} \right| } \cdot \frac{3 ^{x} }{3 ^{ \left|x \right| } }$

2. Originally Posted by achacy
[HTML]\frac{3-3 ^{x}}{ \left|3-3 ^{x} \right| } \cdot \frac{3 ^{x} }{3 ^{ \left|x \right| } }[/HTML]
$\displaystyle f(x) = \frac{3-3 ^{x}}{ \left|3-3 ^{x} \right| } \cdot \frac{3 ^{x} }{3 ^{ \left|x \right| } }$

1. $\displaystyle D = \mathbb{R} \setminus \{1\}$

2.$\displaystyle \dfrac{3-3 ^{x}}{ \left|3-3 ^{x} \right| } = \left\{\begin{array}{r}-1 \ if \ x > 1 \\ 1 \ if \ x < 1 \end{array}\right.$

3. $\displaystyle \dfrac{3^x}{3^{|x|}} = \left\{\begin{array}{r} 1\ if\ x \geq 0 \\ -1\ if\ x<0\end{array}\right.$

Thus you have 3 different intervals:

$\displaystyle f(x)=\left\{\begin{array}{ccr}1 \cdot (-1) & if & x<0 \\ 1 \cdot (+1) & if & 0 \leq x < 1 \\ (-1) \cdot (+1) & if & x>1\end{array} \right.$

I assume that you now can draw the graph.

3. Originally Posted by earboth
[tex]

3.$\displaystyle \dfrac{3^x}{3^{|x|}} = \left\{\begin{array}{r} 1\ if\ x \geq 0 \\ -1\ if\ x<0\end{array}\right.$
But I think if $\displaystyle x<0$ $\displaystyle \frac{3 ^{x} }{ 3^{ \left| x\right| } } =3 ^{2x}$

isn't it so?

4. Originally Posted by achacy
But I think if $\displaystyle x<0$ $\displaystyle \frac{3 ^{x} }{ 3^{ \left| x\right| } } =3 ^{2x}$

isn't it so?
Almost!

First of all: You are right: My result wasn't correct but ...

Let x = -2 then $\displaystyle \dfrac{3 ^{-2} }{ 3^{ \left|-2 \right| } } =\dfrac{\frac1{3 ^2}}{3^{|-2|}} = \dfrac1{3^{2+ 2}}$

In general: $\displaystyle \dfrac{3 ^{x} }{ 3^{ \left| x\right| } } =3 ^{-2|x|}\ if \ x<0$

EDIT: I've attached the graph of the function.