Question

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

(0, 1), (1, 2), (-1, 0) are indicated points on the graph.
options:

• $f(x) < (1 + x)^{\frac{1}{3}}$
• $f(x) \geq (1 + x)^{\frac{1}{3}}$
• $f(x) \leq (1 + x)^{\frac{1}{3}}$
• $f(x) > (1 + x)^{\frac{1}{3}}$

Explanation:

Step1: Analyze the boundary line

The dashed curve suggests the inequality is strict (not including the boundary), so we can eliminate options with $\geq$ or $\leq$. Now we have two options: $f(x) < (1 + x)^{\frac{1}{3}}$ or $f(x) > (1 + x)^{\frac{1}{3}}$.

Step2: Test a point in the shaded region

Take the point $(1, 2)$. Let's compute $(1 + 1)^{\frac{1}{3}} = 2^{\frac{1}{3}} \approx 1.26$. The $y$-value of the shaded region at $x = 1$ is $2$, which is greater than $2^{\frac{1}{3}}$. So $f(x)$ (the shaded region) is greater than the curve $(1 + x)^{\frac{1}{3}}$.

Step3: Confirm the inequality

Since the boundary is dashed (excluding the curve) and the shaded region is above the curve, the correct inequality is $f(x) > (1 + x)^{\frac{1}{3}}$.

Answer:

$f(x) > (1 + x)^{\frac{1}{3}}$ (the last option, assuming the options are ordered as given: the fourth option with $f(x) > (1 + x)^{\frac{1}{3}}$)