Question

1. choose the correct equation or inequality by using the indicated points.

options:
$f(x) > (1 + x)^{\frac{1}{2}}$
$f(x) geq (1 + x)^{\frac{1}{2}}$
$f(x) < (1 + x)^{\frac{1}{2}}$
$f(x) leq (1 + x)^{\frac{1}{2}}$

Explanation:

Step1: Identify boundary curve

The dashed curve is $y=(1+x)^{\frac{1}{2}}$. Dashed lines mean the boundary is not included, so we use $<$ or $>$.

Step2: Test the point (1,2)

Substitute $x=1$ into $y=(1+x)^{\frac{1}{2}}$:
$y=(1+1)^{\frac{1}{2}}=\sqrt{2}\approx1.414$
The point (1,2) is in the shaded region, and $2 > 1.414$, so $f(x) > (1+x)^{\frac{1}{2}}$ holds for this point.

Step3: Verify boundary exclusion

The points $(-1,0)$ and $(0,1)$ are open circles, confirming the boundary is not included, matching the strict inequality $>$.

Answer:

$f(x) > (1 + x)^{\frac{1}{2}}$