Question

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

graph with points (-1, 0), (0, 1), (1, 2)
options:

• $f(x) = 1 - \sqrt3{x}$
• $f(x) = 1 + \sqrt{x}$
• $f(x) = x^3 + 1$
• $f(x) = 1 + \sqrt3{x}$

Explanation:

Step1: Test point (-1, 0)

Substitute \( x = -1 \) into each function:

• For \( f(x)=1 - \sqrt[3]{x} \): \( 1 - \sqrt[3]{-1}=1 - (-1)=2

eq0 \)

• For \( f(x)=1 + \sqrt{x} \): \( \sqrt{-1} \) is undefined (domain issue, \( x\geq0 \) but \( x=-1 \) here)
• For \( f(x)=x^3 + 1 \): \( (-1)^3 + 1=-1 + 1 = 0 \)? Wait, but check another point. At \( x = 1 \), \( 1^3+1 = 2 \), but let's check the next function.
• For \( f(x)=1 + \sqrt[3]{x} \): \( 1 + \sqrt[3]{-1}=1 - 1 = 0 \), which matches \( (-1, 0) \).

Step2: Test point (0, 1)

Substitute \( x = 0 \) into \( f(x)=1 + \sqrt[3]{x} \): \( 1+\sqrt[3]{0}=1 + 0 = 1 \), which matches \( (0, 1) \).

Step3: Test point (1, 2)

Substitute \( x = 1 \) into \( f(x)=1 + \sqrt[3]{x} \): \( 1+\sqrt[3]{1}=1 + 1 = 2 \), which matches \( (1, 2) \).

Answer:

\( f(x) = 1 + \sqrt[3]{x} \) (the last option, e.g., if options are labeled as D: \( f(x) = 1 + \sqrt[3]{x} \), then D. \( f(x) = 1 + \sqrt[3]{x} \))