Question

Which of the following equations correctly describes the function in the table using function notation
0,-5
1,-9
2, -13
3, -17
f(x)=-5x-4
f(x)=-4x-5
f(y) = -4x-5
f(y) =-5x-4

Answer 1

To determine which equation describes the function in the table, we can analyze the given input-output pairs:

- For \( x = 0 \), \( f(0) = -5 \)
- For \( x = 1 \), \( f(1) = -9 \)
- For \( x = 2 \), \( f(2) = -13 \)
- For \( x = 3 \), \( f(3) = -17 \)

Now, we can look at the output values to see how they change as \( x \) increases:

- From \( -5 \) to \( -9 \) is a decrease of \( 4 \).
- From \( -9 \) to \( -13 \) is a decrease of \( 4 \).
- From \( -13 \) to \( -17 \) is a decrease of \( 4 \).

This indicates the function is linear with a slope of \( -4 \).

Now, we can plug into the two options for slope to see which one fits:

### Testing \( f(x) = -4x - 5 \)

- \( f(0) = -4(0) - 5 = -5 \) (correct)
- \( f(1) = -4(1) - 5 = -4 - 5 = -9 \) (correct)
- \( f(2) = -4(2) - 5 = -8 - 5 = -13 \) (correct)
- \( f(3) = -4(3) - 5 = -12 - 5 = -17 \) (correct)

### Testing \( f(x) = -5x - 4 \)

- \( f(0) = -5(0) - 4 = -4 \) (incorrect)
- \( f(1) = -5(1) - 4 = -5 - 4 = -9 \) (correct)
- \( f(2) = -5(2) - 4 = -10 - 4 = -14 \) (incorrect)
- \( f(3) = -5(3) - 4 = -15 - 4 = -19 \) (incorrect)

### Conclusion

The correct equation that describes the function in the table is:

**\( f(x) = -4x - 5 \)**.