Question

g(x) = x² + x + 2
x | h(x)
0 | 0
-1 | -10
3 | 2
2 | -10

Explanation:

Since the problem (presumably related to analyzing functions \(g(x)\) and \(h(x)\) from the table and formula) is in Mathematics, specifically Algebra (dealing with functions, their properties, maybe symmetry or evaluations). Let's assume we need to check symmetry or find something about \(h(x)\).

Step1: Analyze \(h(x)\) table for symmetry

Check if \(h(x)\) is symmetric about a vertical line. For \(x = -1\) and \(x = 2\), \(h(-1)=h(2)= -10\). The midpoint of \(x=-1\) and \(x = 2\) is \(\frac{-1 + 2}{2}=\frac{1}{2}\). For \(x = 0\) and \(x = 3\), midpoint is \(\frac{0+3}{2}=\frac{3}{2}\)? Wait no, wait \(x=0\) and \(x = 3\): \(h(0)=0\), \(h(3)=2\) – not equal. Wait maybe \(h(x)\) is a quadratic? Let's check the vertex. The axis of symmetry for a quadratic is \(x = -\frac{b}{2a}\). If we assume \(h(x)=ax^2+bx + c\), using points \((0,0)\), \((-1,-10)\), \((3,2)\).

Using \((0,0)\): \(c = 0\). So \(h(x)=ax^2+bx\). Plug in \((-1,-10)\): \(a(-1)^2 + b(-1)=a - b=-10\). Plug in \((3,2)\): \(a(3)^2 + b(3)=9a + 3b=2\). Now solve the system:

From first equation: \(a = b - 10\). Substitute into second: \(9(b - 10)+3b=2\) → \(9b - 90 + 3b=2\) → \(12b=92\) → \(b=\frac{92}{12}=\frac{23}{3}\), \(a=\frac{23}{3}-10=\frac{23 - 30}{3}=-\frac{7}{3}\). So \(h(x)=-\frac{7}{3}x^2+\frac{23}{3}x\). Let's check \(x = 2\): \(-\frac{7}{3}(4)+\frac{23}{3}(2)=-\frac{28}{3}+\frac{46}{3}=\frac{18}{3}=6\)? Wait no, but \(h(2)= -10\). So maybe my assumption is wrong. Alternatively, maybe the problem is to evaluate \(g(x)\) at some points or compare with \(h(x)\).

Wait, maybe the problem is to find \(g(x)\) values at \(x\) from the table. Let's compute \(g(0)=0^2 + 0 + 2=2\), \(g(-1)=(-1)^2+(-1)+2=1 -1 +2=2\), \(g(3)=3^2+3 + 2=9 + 3 + 2=14\), \(g(2)=2^2+2 + 2=4 + 2 + 2=8\).

But since the problem is not fully stated, maybe it's about comparing \(g(x)\) and \(h(x)\) at certain \(x\) or finding their properties. If we assume the question is to find which function has a minimum or something, but without the full question, let's assume we need to find the axis of symmetry for \(h(x)\) from the table.

Looking at \(x=-1\) and \(x=2\) have \(h(x)=-10\), so the axis of symmetry is \(x=\frac{-1 + 2}{2}=0.5\) (midpoint). For \(g(x)=x^2 + x + 2\), axis of symmetry is \(x=-\frac{1}{2(1)}=-\frac{1}{2}\).

Answer:

(Assuming the question was about axis of symmetry for \(h(x)\) from table: axis at \(x = \frac{1}{2}\); for \(g(x)\) at \(x=-\frac{1}{2}\)) But since the original question is not fully provided, if we assume the task was to find the axis of symmetry for \(h(x)\) from the table, the answer is \(x=\frac{1}{2}\).