A Function Translation Card for TSIA2 Quantitative Practice

Function questions become easier to audit when every representation is translated into the same three-part sentence: input, rule, output. This guide turns that sentence into a repeatable practice routine for TSIA2-style quantitative work.

The three boxes

Draw three boxes and label them input, rule, and output. When a question gives f(3) = 11, place 3 in the input box and 11 in the output box. The symbol f names the rule; it is not a variable to multiply by 3.

Use this translation sentence:

When the input is 3, the rule named f produces an output of 11.

The same sentence works for other representations:

  • A table row containing x = 3 and y = 11 gives the same input-output pair.
  • A graph containing the point (3, 11) gives the same pair when the horizontal axis is the input and the vertical axis is the output.
  • A verbal rule such as �?odouble the input and add five�?� can be written as f(x) = 2x + 5.

Moving every representation into the same three boxes makes it easier to compare equations, tables, graphs, and descriptions.

Four checks before calculating

1. Name the requested quantity

Is the question asking for an output, an input, the rule, a starting value, a rate of change, or a comparison between two outputs? Write the requested quantity in a short phrase before touching the answer choices.

2. Copy the input exactly

Parentheses may contain a negative number, an expression, or another function value. Treat the entire contents as the input. If the question asks for g(a + 2), substitute a + 2 as one unit rather than substituting only a.

3. Apply the rule once

Substitute the input into every occurrence of the rule�?Ts variable, then simplify with the usual order of operations. Writing the substitution line makes missing parentheses visible.

4. Restate the result

Write a complete statement such as f(-2) = 7, not only the number 7. The notation exposes input-output reversals and makes the final answer easier to compare with the question.

A negative-input diagnostic

Suppose the rule is g(x) = 2x² - 5 and the input is -3. Write:

g(-3) = 2(-3)² - 5

before doing the arithmetic. The parentheses preserve the negative input. Squaring produces 9, so the output is 13.

Writing -3² without substitution parentheses can invite a sign mistake because the exponent applies before the leading negative. The diagnostic rule is simple: every substituted negative value receives parentheses before any calculation begins.

Use an estimate as an independent check. The squared term is nonnegative and large enough that doubling it and subtracting 5 should still produce a positive result. A negative final answer would conflict with that expectation and deserves a recheck.

Reading a function from a graph

For a question such as �?ofind h(4),�?� begin at 4 on the horizontal axis, move vertically until you meet the graph, then move horizontally to read the output axis.

For a question such as �?ofor which input is h(x) = 4,�?� reverse the search:

  1. Begin at output 4 on the vertical axis.
  2. Move horizontally to the graph.
  3. Move down to the horizontal axis to read the input.

Write given input or given output beside the question before tracing. That label prevents the visual direction from being reversed.

If a horizontal line meets the graph more than once, the same output has multiple inputs. Record all of them unless the question limits the domain.

Comparing two representations

When one function is an equation and another is a table or graph, translate both into input-output pairs at the same input value. Do not compare a rate from one representation with a single output from another.

Use a two-line comparison:

f(2) = ___
g(2) = ___

Only after both lines are complete should you compare the values.

If the question asks which function grows faster, compare rates of change rather than isolated outputs. For a linear equation, the coefficient of x supplies the rate. For a table, divide the change in output by the matching change in input. For a graph, choose two readable points and compute rise over run.

If the question asks for an intersection, look for an input where both functions produce the same output. An intersection is a shared input-output pair, not merely two similar-looking values.

A verbal-rule checkpoint

Verbal descriptions often contain an initial value and a repeated change. Separate them before writing an equation.

For example:

A plan costs $12 initially and then $4 for each month.

The starting value is 12, and the rate is 4 per month. If m represents months, the rule is C(m) = 4m + 12.

Check the rule with an input of zero. C(0) = 12, which matches the stated initial cost. The zero-input test is a fast way to distinguish a starting value from a rate.

A short practice cycle

Complete five mixed function items. For every missed or uncertain item, record which checkpoint failed:

  • input and output were reversed;
  • the requested quantity was misread;
  • a negative substitution lost its parentheses;
  • a table or graph was traced in the wrong direction;
  • a value was compared with a rate;
  • one condition, such as a domain restriction, was ignored.

On the next set, focus on only the most common checkpoint. If most errors come from reversing input and output, write the full translation sentence for every item. If errors come from substitution signs, require parentheses around every substituted value. If errors come from comparing unlike quantities, complete the two-line comparison before selecting an answer.

For additional question practice, use the official TSI Practice Test listing on Google Play. Pair each short set with the translation card, then track whether the same error type declines across sessions.

Publisher disclosure: This original educational resource was prepared by a publisher associated with the linked TSI Practice Test app. It is not official TSIA2 testing material and is not affiliated with the Texas Higher Education Coordinating Board or College Board.