# if g(x)= x-2/3x and h(x)=3x+6/4, what is the domain of g(x) * h(x)?

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### 2 Answers

We first multiply the g(x) and h(x).

`g(x) * h(x) = (x - 2)/(3x) * (3x + 6)/4 = ((x - 2)*3(x + 2))/(3x*4) = ((x - 2)(x + 2))/(4x)`

Domain of g(x) is `(- oo, 0)uu (0, oo)`

Domain of h(x) is `( - oo, oo)`

Therefore, **domain of g(x) * h(x) is `(- oo, 0) uu (0, oo)`** ` `

Both g(x) that h(x) are defined on all Reals and so their domain:

`g(x)=x/3` `h(x)=3x+3/2` `f(x)=x^2+x/2=g(x) xx h(x)` is defined to all real too, instead its domain isto be searched:

`f'(x)=2x+1/2`

`f'(x)=0` `rArr` `x=-1/4`

`f''(x)= 2>0` is a min point.

`m=f(-1/4)= -1/8`

so its domani is `-1/8 <= x < oo`

Blue line g(x), green line h(x), red line f(x)= g(x)h(x)