Reading Functions from Tables: Patterns, Inputs, and Output Behavior

1) Reading a Function Directly from a Table

A table can describe a function by listing specific input values and their corresponding outputs. Your job is to read the table like a “lookup tool”: match an input to its output.

Identify the input column and output column

Most tables label columns with something like x (input) and f(x), y, or “output” (output). If labels are missing, look for clues: the input column often contains the values being “fed into” the rule, and the output column contains the results.

Use function notation with table lookups

To find f(3), locate the row where the input is 3, then read the output in the same row.

• Input 3 corresponds to output 4, so f(3) = 4.
• Input 0 corresponds to output 1, so f(0) = 1.

Step-by-step: answering “Find f(a)” from a table

1. Find the input value a in the input column.
2. Move across that row to the output column.
3. Write the result as f(a) = (output).

Important: If the input value is not listed, you cannot read f(a) exactly from the table unless you are told a pattern or rule.

2) Detecting Patterns: Constant Differences and Second Differences

Sometimes a table is built from a pattern. Recognizing the pattern helps you predict values between or beyond the listed rows (when it is reasonable to do so and when the pattern is stated or strongly suggested).

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Constant first difference (intro to linear behavior)

If the inputs increase by a constant amount (often by 1), and the outputs change by a constant amount each step, the table shows a constant first difference.

Because the first differences are constant (+3 each time), the outputs follow a linear pattern.

Constant second difference (intro to quadratic behavior)

If the first differences are not constant, check the second differences (the differences of the differences). When the second differences are constant (again assuming equal input steps), the table often shows quadratic behavior.

The first differences change, but the second differences are constant (+2), suggesting a quadratic pattern.

Checklist for pattern detection (when inputs step evenly)

• Step 1: Confirm the input values increase by a constant amount (like +1 each row).
• Step 2: Compute first differences of outputs.
• Step 3: If first differences are constant, treat it as linear behavior.
• Step 4: If not, compute second differences; if those are constant, treat it as quadratic behavior.

Note: A table can match a pattern for the listed points but still not represent the full function rule. Tables often show only part of a function.

3) Estimating Missing Values When a Pattern Is Stated

Sometimes a table has blanks. If you are told the pattern type (for example, “the outputs increase by 4 each time” or “the second differences are constant”), you can fill in missing outputs.

Example A: Fill missing values using a constant first difference

Suppose the table follows a constant first difference of +4 in the outputs as x increases by 1.

Step-by-step:

1. From x=1 to x=2, output goes from 7 to 11, which is +4.
2. Apply the same change from x=2 to x=3: 11 + 4 = 15.
3. Check: from x=3 to x=4 should be +4, and 15 + 4 = 19, which matches.

So g(3) = 15.

Example B: Fill missing values using constant second differences

Suppose x increases by 1 each row and the table is known to have constant second difference +2.

Step-by-step idea: build the difference table.

• First difference from x=0 to x=1 is 5 - 2 = 3.
• Let the missing value be h(2)=A. Then first differences are: A - 5 (from 1 to 2) and 17 - A (from 2 to 3).
• Second differences must be +2: (A - 5) - 3 = 2.
• Solve: A - 8 = 2 so A = 10.
• Check the next second difference: (17 - 10) - (10 - 5) = 7 - 5 = 2, works.

So h(2)=10.

When you should (and should not) fill in blanks

• Do fill in missing values when a pattern is stated (constant difference, constant second difference) or when the table clearly shows it and the problem asks you to extend it.
• Do not assume a pattern if the problem does not give one and the table is short or irregular; many different functions can fit the same few points.

4) Recognizing Domain and Range from Listed Values

From a table, you can read the domain and range of the listed data directly.

• The domain (from the table) is the set of input values shown in the input column.
• The range (from the table) is the set of output values shown in the output column (without repeating duplicates).

From this table:

• Domain (listed): {-2, 0, 2, 5}
• Range (listed): {1, 3, 10} (note that 3 appears twice in the table but is listed once in the set)

Reminder: This is the domain and range of the table entries, not necessarily the full domain and range of the function beyond the table.

Practice: Table Reading + Pattern Recognition

A. Exact table lookups

• 1) Find f(-1).
• 2) Find f(2).
• 3) Is f(1) determined by this table? Explain using the table entries.

B. Identify input vs output and write statements

• 4) Treat T as a function of t. Write T(2) = ?
• 5) What does T(4) = 24 mean in words?

C. Constant first differences (linear pattern)

• 6) Compute the first differences of g(x).
• 7) Use the pattern to predict g(5).

D. Constant second differences (quadratic pattern)

• 8) Compute first differences and second differences.
• 9) Based on the differences, what type of pattern does the table suggest (linear or quadratic)?

E. Fill in missing values when the pattern is stated

For each table, assume x increases by 1 each row.

• 10) If the outputs follow a constant first difference, find m(4).

• 11) If the table has constant second difference +3, find n(3).

F. Domain and range from listed values

• 12) List the domain (from the table).
• 13) List the range (from the table), without repeats.
• 14) Does the table show enough information to know q(1)? Why or why not?