Question

the graph of a function f(x) passes through the following points:
(0,-2),(1,0),(-1,-4)
which of the following could be f(x)?
a)f(x)=2x-2
b)f(x)=2x^2-2
c)f(x)=-2x-2
d)f(x)=2√x-2

Answer 1

To determine which function \( f(x) \) passes through the points \((0, -2)\), \((1, 0)\), and \((-1, -4)\), we can evaluate each of the provided options at these specific \( x \) values.

### a) \( f(x) = 2x - 2 \)

- For \( x = 0 \):
\[
f(0) = 2(0) - 2 = -2 \quad \text{(matches)}
\]
- For \( x = 1 \):
\[
f(1) = 2(1) - 2 = 0 \quad \text{(matches)}
\]
- For \( x = -1 \):
\[
f(-1) = 2(-1) - 2 = -2 - 2 = -4 \quad \text{(matches)}
\]

This function matches all three points.

### b) \( f(x) = 2x^2 - 2 \)

- For \( x = 0 \):
\[
f(0) = 2(0)^2 - 2 = -2 \quad \text{(matches)}
\]
- For \( x = 1 \):
\[
f(1) = 2(1)^2 - 2 = 2 - 2 = 0 \quad \text{(matches)}
\]
- For \( x = -1 \):
\[
f(-1) = 2(-1)^2 - 2 = 2 - 2 = 0 \quad \text{(does NOT match)}
\]

This function does NOT match all three points.

### c) \( f(x) = -2x - 2 \)

- For \( x = 0 \):
\[
f(0) = -2(0) - 2 = -2 \quad \text{(matches)}
\]
- For \( x = 1 \):
\[
f(1) = -2(1) - 2 = -2 - 2 = -4 \quad \text{(does NOT match)}
\]
- For \( x = -1 \):
\[
f(-1) = -2(-1) - 2 = 2 - 2 = 0 \quad \text{(does NOT match)}
\]

This function does NOT match all three points.

### d) \( f(x) = 2\sqrt{x} - 2 \)

- For \( x = 0 \):
\[
f(0) = 2\sqrt{0} - 2 = -2 \quad \text{(matches)}
\]
- For \( x = 1 \):
\[
f(1) = 2\sqrt{1} - 2 = 2 - 2 = 0 \quad \text{(matches)}
\]
- For \( x = -1 \):
\[
f(-1) \quad \text{(undefined as you can't take the square root of a negative number)}
\]

This function does NOT match all three points.

### Conclusion

The only function that matches all three points is:

**a)** \( f(x) = 2x - 2 \)