Question

1. choose the correct equation or inequality by using the indicated points. the chart shows a graph with points (0,0), (1,1), (4,2). the options are: $f(x) = x$, $f(x) = x^{\frac{1}{2}}$, $f(x) = x^2$, $f(x) = sqrt3{x}$

Explanation:

Step1: Test point (0,0) in all functions

• For $f(x)=x$: $f(0)=0$, works.
• For $f(x)=x^\frac{1}{2}$: $f(0)=0^\frac{1}{2}=0$, works.
• For $f(x)=x^2$: $f(0)=0^2=0$, works.
• For $f(x)=\sqrt[3]{x}$: $f(0)=\sqrt[3]{0}=0$, works.

Step2: Test point (1,1) in all functions

• For $f(x)=x$: $f(1)=1$, works.
• For $f(x)=x^\frac{1}{2}$: $f(1)=1^\frac{1}{2}=1$, works.
• For $f(x)=x^2$: $f(1)=1^2=1$, works.
• For $f(x)=\sqrt[3]{x}$: $f(1)=\sqrt[3]{1}=1$, works.

Step3: Test point (4,2) in all functions

• For $f(x)=x$: $f(4)=4

eq 2$, fails.

• For $f(x)=x^\frac{1}{2}$: $f(4)=4^\frac{1}{2}=\sqrt{4}=2$, works.
• For $f(x)=x^2$: $f(4)=4^2=16

eq 2$, fails.

• For $f(x)=\sqrt[3]{x}$: $f(4)=\sqrt[3]{4} \approx 1.587

eq 2$, fails.

Answer:

$\boldsymbol{f(x) = x^\frac{1}{2}}$