Question

F(x)=|1/3x| translation 2 units to the left

Answer 1

`f(x)=\left|(1)/(3)x\right|\xrightarrow{T_{\vec{a}= <-2;\ 0 >}}g(x)=f(x+2)=\left|(1)/(3)(x+2)\right|=\left|(1)/(3)x+(2)/(3)\right|`

Look at the picture.

Answer 2

`f(x)=(1)/(3)|(x+2)|`

Explanation:

We have been given a function
`f(x)=|(1)/(3)x|`. We are asked to translate the function to 2 units to the left.

We know that an absolute function is form
`y=a|x-h|+k`, where, (h,k) is vertex.

Let us recall translation rules.

`f(x)\rightarrow f(x-a)=\text{Graph shifted to right by a units}`

`f(x)\rightarrow f(x+a)=\text{Graph shifted to left by a units}`

`f(x)\rightarrow f(x)+a=\text{Graph shifted upwards by a units}`

`f(x)\rightarrow f(x)-a=\text{Graph shifted downwards by a units}`

Absolute value of 1/3 will be 1/3.

`f(x)=(1)/(3)|x|`

Therefore, our required function would be
`f(x)=(1)/(3)|x+2|`.