Question

The table below represents a function.

x y
0 1
2 5
5 26
7 50
Which equation makes the table true?

f( x) = x^2 - 1
f( x) = x^2 + 1
f( x) = 2 x - 1
f( x) = 2 x + 1

Answer 1

To determine which equation represents the function given in the table, we can evaluate each equation at the provided \( x \) values and see which one yields the correct \( y \) values.

The equations to test are:

1. \( f(x) = x^2 - 1 \)
2. \( f(x) = x^2 + 1 \)
3. \( f(x) = 2x - 1 \)
4. \( f(x) = 2x + 1 \)

Let's calculate \( y \) for each \( x \) given in the table:

1. **Testing \( f(x) = x^2 - 1 \)**:
- \( f(0) = 0^2 - 1 = -1 \) (not correct)
- \( f(2) = 2^2 - 1 = 4 - 1 = 3 \) (not correct)
- \( f(5) = 5^2 - 1 = 25 - 1 = 24 \) (not correct)
- \( f(7) = 7^2 - 1 = 49 - 1 = 48 \) (not correct)

2. **Testing \( f(x) = x^2 + 1 \)**:
- \( f(0) = 0^2 + 1 = 0 + 1 = 1 \) (correct)
- \( f(2) = 2^2 + 1 = 4 + 1 = 5 \) (correct)
- \( f(5) = 5^2 + 1 = 25 + 1 = 26 \) (correct)
- \( f(7) = 7^2 + 1 = 49 + 1 = 50 \) (correct)

3. **Testing \( f(x) = 2x - 1 \)**:
- \( f(0) = 2(0) - 1 = -1 \) (not correct)
- \( f(2) = 2(2) - 1 = 4 - 1 = 3 \) (not correct)
- \( f(5) = 2(5) - 1 = 10 - 1 = 9 \) (not correct)
- \( f(7) = 2(7) - 1 = 14 - 1 = 13 \) (not correct)

4. **Testing \( f(x) = 2x + 1 \)**:
- \( f(0) = 2(0) + 1 = 1 \) (correct)
- \( f(2) = 2(2) + 1 = 4 + 1 = 5 \) (correct)
- \( f(5) = 2(5) + 1 = 10 + 1 = 11 \) (not correct)
- \( f(7) = 2(7) + 1 = 14 + 1 = 15 \) (not correct)

Based on these evaluations, the function that accurately represents the relationship in the table is:

**\( f(x) = x^2 + 1 \)**.