Question

the graph of a function ( f(x) ) passes through the following points: ( (0, -2), (1, 0), (-1, 0) ) which of the following could be ( f(x) )? ( \bigcirc f(x) = 2x - 2 ) ( \bigcirc f(x) = 2sqrt{x} - 2 ) ( \bigcirc f(x) = 2x^2 - 2 ) ( \bigcirc f(x) = -2x - 2 )

Explanation:

Step1: Test point (0, -2) on all options

• For $f(x)=2x-2$: $f(0)=2(0)-2=-2$ ✔️
• For $f(x)=2\sqrt{x}-2$: $f(0)=2\sqrt{0}-2=-2$ ✔️
• For $f(x)=2x^2-2$: $f(0)=2(0)^2-2=-2$ ✔️
• For $f(x)=-2x-2$: $f(0)=-2(0)-2=-2$ ✔️

Step2: Test point (1, 0) on all options

• For $f(x)=2x-2$: $f(1)=2(1)-2=0$ ✔️
• For $f(x)=2\sqrt{x}-2$: $f(1)=2\sqrt{1}-2=0$ ✔️
• For $f(x)=2x^2-2$: $f(1)=2(1)^2-2=0$ ✔️
• For $f(x)=-2x-2$: $f(1)=-2(1)-2=-4

eq0$ ❌

Step3: Test point (-1, 0) on remaining options

• For $f(x)=2x-2$: $f(-1)=2(-1)-2=-4

eq0$ ❌

• For $f(x)=2\sqrt{x}-2$: $\sqrt{-1}$ is undefined ❌
• For $f(x)=2x^2-2$: $f(-1)=2(-1)^2-2=0$ ✔️

Answer:

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