Answer the questions below to determine what kind of function is depicted in the table below.

Question

xx 11 22 33 44 55
f, of, xf(x) 1717 2424 3131 3838 4545
Answer
Attempt 1 out of 2

This function is
because

Answer 1

To determine the type of function represented in the table, we can observe the values of \( x \) and their corresponding values of \( f(x) \):

• \( f(11) = 1717 \)
• \( f(22) = 2424 \)
• \( f(33) = 3131 \)
• \( f(44) = 3838 \)
• \( f(55) = 4545 \)

We can look for a pattern in the values of \( f(x) \).

If we rewrite \( f(x) \) in terms of \( x \):

1. For \( x = 11 \):

   • \( f(11) = 1717 = 11 \times 1 + 1716 \)

2. For \( x = 22 \):

   • \( f(22) = 2424 = 22 \times 1 + 2402 \)

3. For \( x = 33 \):

   • \( f(33) = 3131 = 33 \times 1 + 3098 \)

4. For \( x = 44 \):

   • \( f(44) = 3838 = 44 \times 1 + 3794 \)

5. For \( x = 55 \):

   • \( f(55) = 4545 = 55 \times 1 + 4490 \)

It appears that \( f(x) = 100x - 100 \). Plugging in values of \( x \):

• For \( x = 11 \): \( f(11) = 100(11) - 100 = 1100 - 100 = 1000 \) (incorrect)
• For x = 22: \( f(22) = 2200 - 100 = 2900 \) (incorrect)
• For x = 33: \( f(33) = 3300 - 100 = 3200 \) (incorrect)
• For x = 44: \( f(44) = 4400 - 100 = 4300 \) (incorrect)
• For x = 55: \( f(55) = 5500 - 100 = 5400 \) (incorrect)

So instead we realize we should focus on differences.

Calculating the difference between sequential values of \( f(x) \):

• \( 2424 - 1717 = 707 \)
• \( 3131 - 2424 = 707 \)
• \( 3838 - 3131 = 707 \)
• \( 4545 - 3838 = 707 \)

The difference between consecutive \( f(x) \) values is constant (always 707), which indicates that \( f(x) \) is a linear function.

Thus, we conclude:

This function is linear because the differences between consecutive values of \( f(x) \) are constant.