Given: g(x)=(1)/(x+2) and h(x)=3x Check all restrictions on the domain of gOh xneq 0 III xneq -2 xneq -(2)/(3)

To find the restrictions on the domain of the composite function gOh, we need to consider the restrictions on the individual functions g(x) and h(x).<br /><br />For the function g(x) = 1/(x+2), the denominator cannot be equal to zero. Therefore, the restriction on the domain of g(x) is x ≠ -2.<br /><br />For the function h(x) = 3x, there are no restrictions on the domain since it is a linear function.<br /><br />Now, let's consider the composite function gOh, which means we need to substitute h(x) into g(x). <br /><br />g(h(x)) = g(3x) = 1/(3x+2)<br /><br />In this case, the denominator cannot be equal to zero. Therefore, the restriction on the domain of gOh is 3x + 2 ≠ 0, which simplifies to x ≠ -2/3.<br /><br />So, the restrictions on the domain of gOh are:<br />- x ≠ -2<br />- x ≠ -2/3<br /><br />Therefore, the correct answer is:<br />III $x\neq -\frac {2}{3}$