Answer :
To determine which function [tex]\( f(x) \)[/tex] matches the given points [tex]\((0, -2)\)[/tex], [tex]\((1, 0)\)[/tex], and [tex]\((-1, -4)\)[/tex], we will evaluate each given function at these points and check for consistency.
### Candidate functions:
1. [tex]\( f(x) = 2x^2 - 2 \)[/tex]
2. [tex]\( f(x) = 2\sqrt{x} - 2 \)[/tex]
3. [tex]\( f(x) = 2x - 2 \)[/tex]
4. [tex]\( f(x) = -2x - 2 \)[/tex]
### Evaluating each function:
#### Function 1: [tex]\( f(x) = 2x^2 - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2(0)^2 - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2(1)^2 - 2 = 0 \][/tex]
This matches the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2(-1)^2 - 2 = 0 \][/tex]
This does not match the point [tex]\((-1, -4)\)[/tex].
So, this function does not work.
#### Function 2: [tex]\( f(x) = 2\sqrt{x} - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2\sqrt{0} - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2\sqrt{1} - 2 = 0 \][/tex]
This matches the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ \text{We cannot evaluate }\sqrt{-1}\text{ since it is not a real number.} \][/tex]
Thus, this function is invalid for all the given points.
#### Function 3: [tex]\( f(x) = 2x - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2(0) - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2(1) - 2 = 0 \][/tex]
This matches the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2(-1) - 2 = -4 \][/tex]
This matches the point [tex]\((-1, -4)\)[/tex].
So, this function fits all given points.
#### Function 4: [tex]\( f(x) = -2x - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0) - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -2(1) - 2 = -4 \][/tex]
This does not match the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -2(-1) - 2 = 0 \][/tex]
This does not match the point [tex]\((-1, -4)\)[/tex].
So, this function does not work.
### Conclusion:
After evaluating each candidate function, the only function that satisfies all given points is:
[tex]\[ f(x) = 2x - 2 \][/tex]
Thus, the correct function is [tex]\(\boxed{2x - 2}\)[/tex].
### Candidate functions:
1. [tex]\( f(x) = 2x^2 - 2 \)[/tex]
2. [tex]\( f(x) = 2\sqrt{x} - 2 \)[/tex]
3. [tex]\( f(x) = 2x - 2 \)[/tex]
4. [tex]\( f(x) = -2x - 2 \)[/tex]
### Evaluating each function:
#### Function 1: [tex]\( f(x) = 2x^2 - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2(0)^2 - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2(1)^2 - 2 = 0 \][/tex]
This matches the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2(-1)^2 - 2 = 0 \][/tex]
This does not match the point [tex]\((-1, -4)\)[/tex].
So, this function does not work.
#### Function 2: [tex]\( f(x) = 2\sqrt{x} - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2\sqrt{0} - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2\sqrt{1} - 2 = 0 \][/tex]
This matches the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ \text{We cannot evaluate }\sqrt{-1}\text{ since it is not a real number.} \][/tex]
Thus, this function is invalid for all the given points.
#### Function 3: [tex]\( f(x) = 2x - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2(0) - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2(1) - 2 = 0 \][/tex]
This matches the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2(-1) - 2 = -4 \][/tex]
This matches the point [tex]\((-1, -4)\)[/tex].
So, this function fits all given points.
#### Function 4: [tex]\( f(x) = -2x - 2 \)[/tex]
- At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -2(0) - 2 = -2 \][/tex]
This matches the point [tex]\((0, -2)\)[/tex].
- At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = -2(1) - 2 = -4 \][/tex]
This does not match the point [tex]\((1, 0)\)[/tex].
- At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = -2(-1) - 2 = 0 \][/tex]
This does not match the point [tex]\((-1, -4)\)[/tex].
So, this function does not work.
### Conclusion:
After evaluating each candidate function, the only function that satisfies all given points is:
[tex]\[ f(x) = 2x - 2 \][/tex]
Thus, the correct function is [tex]\(\boxed{2x - 2}\)[/tex].
