Think of a train timetable. If you know the train, you look across to find its arrival time. If you know the arrival time, you search for every train arriving then. A function table works the same way. You can start with an input and find its output, or start with an output and find all matching inputs. A graph is another picture of those same pairs. Its horizontal position shows the input, and its vertical position shows the output. Read the question first so you know which direction to search. You do not need a formula when the table or graph supplies the answer.
Evaluation goes from one input to its output; solving searches every copy of a given output.
Reminder
Evaluate versus solve. In g(3), 3 is the given input. In g(n) = 6, 6 is the given output. These are different starting points.
Squaring a negative. Keep parentheses: (−2)2 = 4. The expression −22 means −4.
Why it works. Each column of an input and output table keeps a pair together. Moving down a column therefore preserves the connection between the two values. On a graph, a point's two coordinates preserve that same connection: (3, 7) says input 3 gives output 7. Fixing an input selects one output because the relation is a function. Fixing an output can select several inputs because different inputs may share an output.
RuleEvaluate g(3): input 3 is given; read its output. Solve g(n) = 6: output 6 is given; find every input paired with it. A point (x, y) on a function graph means y = f(x).
The same idea, five ways
Say it
g of three is seven. g of n equals six asks which n values work.
Write it
Evaluation begins with an input. Solving begins with an output and collects all matching inputs.
In math
g(3) = 7
(3, 7)
g(n) = 6 gives n = 2 or n = 4
Points (2, 6) and (4, 6) share height 6.
Like
A train name finds its arrival time; an arrival time can match several trains.
See itEvaluation goes from one input to its output; solving searches every copy of a given output.
The same idea, other ways
As a timetable
Given a train name, look for one arrival time. Given an arrival time, collect every train name next to it. Evaluation and solving reverse the search.
As paired coordinates
The table column with input 3 and output 7 becomes the graph point (3, 7). You can read that same pair from either direction.
Evaluation follows the input to its paired output.
.1Table lookup in both directions
A table can work like a contact list. The input can be a name, and you look beside that name for the attached information. This table gives a pet label as input and memory span in hours as output. Call its rule M. Find Gold in the input row to evaluate M(Gold). Search backward from an hour value to find its pet label. The short table labels are Pup for puppy, Dog for adult dog, Gold for goldfish, and Beta for beta fish. Cat is unchanged.
M has the listed pet labels as its domain and the listed hour values as its range.
M(Gold) = 2160 hours in this table. Gold is the goldfish label.
A table can define the entire function or show selected values. Read the statement before assuming missing inputs are allowed.
Reason: A function associates an input with an output; it does not require the input to be a number or the rule to be a formula. The paired entries already tell you the rule for the listed labels. Looking up an unlisted label would need more information. Changing the direction of the search does not change any pair, so checking an answer means reading its column in the other direction.
Memory span in hours. Pup = puppy, Dog = adult dog, Gold = goldfish, and Beta = beta fish.
Reminder
Function notation. M(Cat) means use Cat as the input to M. It does not mean multiplication.
The same idea, five ways
Say it
M of Gold equals two thousand one hundred sixty hours.
Write it
The goldfish input label Gold has an output of 2160 hours.
In math
M(Gold) = 2160
(Gold, 2160)
M(pet) = 3600 gives pet = Beta
Like
Look up a name in a contact list, then read the entry attached to that name.
See itMemory span in hours. Pup = puppy, Dog = adult dog, Gold = goldfish, and Beta = beta fish.
Worked exampleRead a label, then reverse the search
Find M(Gold), the memory span paired with the goldfish label Gold. Then solve M(pet) = 3600, meaning find every pet label paired with 3600 hours.
Memory span in hours. Pup = puppy, Dog = adult dog, Gold = goldfish, and Beta = beta fish.
What it asks. Find the number attached to Gold, then the pet label attached to 3600 hours.
Plan. Read down from Gold for evaluation and up from 3600 for solving.
Gold is the table label for goldfish.
The hour value 3600 is in the last output cell.
Locate Gold in the input row and read 2160 directly below it.The output in that column is M(Gold), the memory span for goldfish.
Locate 3600 in the output row and read Beta directly above it.Solving reverses the lookup, and no other output cell holds 3600.
Answer
M(Gold) = 2160 hours.
M(pet) = 3600 gives pet = Beta, the beta fish label.
Check The Gold column contains 2160 hours. The Beta column contains 3600 hours. Read the same two columns in reverse to confirm each answer.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: M(Hamster) = 16 because a hamster seems similar to a cat.
Hamster is not a listed input, and a resemblance supplies no function value.
✓ Instead: The table does not give M(Hamster).
Tips and tricks
Keep the output unit: pet label goes in, hours come out.
If a pet label is not listed, the table gives no value for it.
As a contact list: A person's name can locate a phone number without an arithmetic formula. A pet label can locate its hour value the same way.
As arrows: Each label points to its paired number. To solve for an output, trace every arrow ending at that number back to its label.
.2Graph reading in both directions
Think of a building directory on a map. Start with an input address along the horizontal axis, move straight up or down until you reach the curve, then read its height as the output. Starting with an output reverses the search: hold that height and look across for every point at that level. The curve s(x) = x2 includes the points between the marked samples because its formula defines them too. Separate table dots alone do not tell you to draw a connecting curve.
The graph of a function is its collection of input and output points (x, f(x)).
For the curve s(x) = x2, each input is squared to find its height.
Evaluate by choosing x first. Solve by choosing y first. A height below this square curve has no matching point.
Reason: All points directly above one horizontal position have the same input. That makes a vertical search useful when the input is given. All points at the same height have the same output. That makes a horizontal search useful when the output is given. Joining separate data dots would claim answers for new inputs. Only a curve or rule gives permission to include those extra points.
This square curve illustrates one evaluation and two inputs sharing height 9.
Reminder
Coordinates. In (−3, 9), go 3 left and 9 up. The order is horizontal first, vertical second.
The same idea, five ways
Say it
s of two equals four. s of x equals nine asks which x values reach nine.
Write it
The curve height gives an output; the horizontal addresses at a given height give its input solutions.
In math
s(2) = 4
(2, 4)
s(x) = 9 gives x = −3 or x = 3
Like
An address finds one shelf height; a shelf height may occur at several addresses.
See itThis square curve illustrates one evaluation and two inputs sharing height 9.
Worked exampleA square curve gives one output and two inputs
For the graph s(x) = x2, evaluate s(2), meaning read the height at input 2. Then solve s(x) = 9, meaning find every input at height 9.
Height 9 meets the square curve twice, so collect both inputs.
What it asks. Read one output at input 2, then all inputs at output height 9.
Plan. Move vertically from x = 2 to the curve. Then follow the horizontal dashed level at height 9 to both contacts.
The point (2, 4) carries input 2 and output 4.
Both points (−3, 9) and (3, 9) have height 9.
Start at 2 on the horizontal axis and move up to the point (2, 4). Read s(2) = 4.The point's vertical coordinate is the output for its horizontal coordinate.
Start at height 9 on the vertical axis and look across to both curve points (−3, 9) and (3, 9).Both points have the requested output 9.
Read their horizontal coordinates and write x = −3 or x = 3.Solving asks for the inputs, rather than the common height.
Answer
s(2) = 4.
For s(x) = 9:
x = −3.
x = 3.
Check Use the rule independently of the picture: 22 = 4, (−3)2 = 9, and 32 = 9.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: s(x) = 9 has the answer 9.
9 is the output already given; the question asks which inputs reach it.
✓ Instead: Read the horizontal coordinates: x = −3 or x = 3.
Tips and tricks
Write input or output beside the given value before touching the graph.
A few marked points on a curve are samples. Separate data dots without a curve do not authorize joining them.
As a map address: For s(2), go to horizontal position 2 and read the curve's height 4. The address (2, 4) contains the answer.
As a level shelf: For s(x) = 9, slide a level shelf across height 9. It touches this curve at horizontal positions −3 and 3.
.3Domain and range from a graph
Imagine shining a lamp straight down onto a curve. Its shadow on the horizontal axis shows every input position the graph covers: the domain. Now shine the lamp from the side. The shadow on the vertical axis shows every output height it reaches: the range. Domain is read left to right. Range is read bottom to top. The drawn window shows a portion of an unending curve, so use its continuing direction together with the rule to decide whether a shadow continues forever.
Domain: the graph’s horizontal input coverage. Read its shadow on the x-axis.
Range: the graph’s vertical output coverage. Read its shadow on the y-axis.
For x, both shadows start at the included zero and continue in the positive direction.
For x2, the bowl reaches left and right forever, so domain is all real numbers. Its heights are zero or positive, so range is [0, ∞).
The input shadow begins at x = 0 and goes right. The output shadow begins at y = 0 and goes up.
Reminder
Interval endpoints. [0, ∞) includes zero. The parenthesis at infinity means there is no last point.
The same idea, five ways
Say it
Domain is left to right. Range is bottom to top.
Write it
Inputs make the horizontal shadow; outputs make the vertical shadow.
In math
For y = x: domain [0, ∞), range [0, ∞).
For y = x2: domain (−∞, ∞), range [0, ∞).
Like
Two lamps cast two different shadows of the same curve.
See itThe input shadow begins at x = 0 and goes right. The output shadow begins at y = 0 and goes up.
Worked exampleRead the square-root graph’s two shadows
Use the graph y = x to find its domain and range.
The input shadow begins at x = 0 and goes right. The output shadow begins at y = 0 and goes up.
What it asks. Find every input position and every output height covered by the graph.
Plan. Project the graph onto the x-axis for domain and onto the y-axis for range. Read included endpoints separately.
Domain is read left to right.
Range is read bottom to top.
Read left to right: the graph starts at input 0 and continues right forever.Domain collects the horizontal input positions covered by the curve.
Write domain [0, ∞).The starting point (0, 0) includes input 0, and the curve has no last input.
Read bottom to top: the lowest height is 0, and the graph continues upward forever.Range collects the vertical output heights the curve reaches.
Write range [0, ∞).Output 0 is included, and no upper height is the last output.
Answer
Domain: [0, ∞).
Range: [0, ∞).
Check Check the rule: negative inputs have no real square root, and a root output is nonnegative. Any output y ≥ 0 is reached by input y2.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The square graph stays above the x-axis, so its domain is [0, ∞).
Above the axis describes output heights. The bowl still has points at negative horizontal inputs, such as (−2, 4).
✓ Instead: For x2, domain is (−∞, ∞); range is [0, ∞).
Tips and tricks
Memory cue: D before R, I before O, x before y. Domain is input x; range is output y.
Check whether a finite endpoint is included before choosing its bracket.
With shadows: Drop the graph’s points onto the x-axis for domain. Move them sideways onto the y-axis for range.
With a point address: The point (4, 2) contributes input 4 to the domain and output 2 to the range. Collect every graph point’s first or second coordinate.
Strategy: step by step
Translate the question into either given input, find output, or given output, find inputs.
In a table, stay in the same column when moving between its input and output rows.
On a graph, locate the given input on the horizontal axis or the given output on the vertical axis.
For evaluation, move vertically to the graph and read the height. For solving, search across the given height and read every horizontal position.
Check each answer against its original column or point. Do not invent values between isolated data points.
Strategy
Read a table or graph in the requested direction
1
Is an input given, as in g(3)?
YesEvaluate: keep that input and find its paired output.
NoAn output is given, so solve by finding every input producing it.
↓
2
Is the information a table?
YesRead down from the input to evaluate; read up from every matching output to solve.
NoOn a graph, move vertically from the input to the curve to evaluate. For solving, search across the given output height.
↓
3
Does the given input or output have a listed column or graph contact?
YesRead the paired coordinate. For solving, keep every matching input.
NoFor a complete function, there is no matching value there. For a sample table, a missing value needs more information.
Translate the question into either given input, find output, or given output, find inputs.
In a table, stay in the same column when moving between its input and output rows.
On a graph, locate the given input on the horizontal axis or the given output on the vertical axis.
For evaluation, move vertically to the graph and read the height. For solving, search across the given height and read every horizontal position.
Check each answer against its original column or point. Do not invent values between isolated data points.
Worked exampleRead the g table in both directions
Evaluate g(3), meaning find the output for input 3. Then solve g(n) = 6, meaning find every input whose output is 6. Use the table in the picture.
Read down under 3 to evaluate; read up from both 6s to solve.
What it asks. Find the output belonging to input 3, then find every input belonging to output 6.
Plan. Read down the highlighted input-3 column. Then find both highlighted 6s in the output row and read up their columns.
Stay directly below 3 to evaluate g(3).
For g(n) = 6, the two 6s belong to inputs 2 and 4.
Find 3 in the input row. Read the output 7 directly below it, so g(3) = 7.The column under 3 holds the output paired with input 3.
Find both 6s in the output row. Follow their columns upward to inputs 2 and 4.Solving starts with the given output and must include every matching column.
Write n = 2 or n = 4.Both inputs produce 6, and the other three columns produce different outputs.
Answer
g(3) = 7.
For g(n) = 6:
n = 2.
n = 4.
Check Read downward again: the column under 2 gives 6 and the column under 4 gives 6. The column under 3 gives 7, so 3 is not a solution of g(n) = 6.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: evaluate one table input
Evaluate g(1), meaning find the output under input 1 in the table.
One highlighted input identifies one paired output.
What it asks. Find the output in the column for input 1.
Plan. Locate input 1 and read directly down.
The first column begins with input 1.
Its lower cell is 8.
Find 1 in the input row and read the 8 below it.Evaluation follows one known input to its paired output.
Answer
g(1) = 8.
Check The first column pairs 1 with 8, so the ordered pair is (1, 8).
Rung 2Rung 2: solve for all table matches
Solve g(n) = 6, meaning find every input whose paired output in the table is 6.
Both output matches must contribute their inputs.
What it asks. Find all inputs that give output 6.
Plan. Find every 6 in the output row, then follow each column up.
There are two 6s.
Their input cells are 2 and 4.
Find both 6s in the output row.Stopping at the first match could lose a solution.
Read inputs 2 and 4 above those cells.The solution values are the inputs paired with the requested output.
Answer
n = 2.
n = 4.
Check Reading downward from each answer gives g(2) = 6 and g(4) = 6.
Rung 3Rung 3: evaluate a graph input
For the curve s(x) = x2, evaluate s(−2), meaning read the output height at input −2.
Input −2 lies left of zero, but its output lies above zero.
What it asks. Read the square curve’s height at input −2.
Plan. Keep x = −2 while moving vertically to the curve, then read y.
−2 is left of the origin.
The point there is (−2, 4).
Start at −2 on the horizontal axis and move vertically to the curve at (−2, 4).The input stays −2 during a vertical move.
Read the height 4, so s(−2) = 4.The vertical coordinate is the output.
Answer
s(−2) = 4.
Check The rule gives (−2)2 = (−2) × (−2) = 4, agreeing with the graph.
Rung 4Rung 4: two graph solutions, then no solutions
On the curve s(x) = x2, solve s(x) = 4 and s(x) = −1. Each question gives an output height and asks for every input at that height.
Height 4 cuts the curve in two places.Height −1 never touches the curve.
What it asks. Find every input at height 4, then every input at height −1.
Plan. Inspect each dashed horizontal level and read the x-coordinate of every curve contact.
Height 4 has two contacts.
The square curve never goes below zero.
At height 4, find curve points (−2, 4) and (2, 4). Read x = −2 or x = 2.Both points have output 4.
At height −1, find no point on the curve. Write no real solution.A square is never negative, so this curve never reaches a height below zero.
Answer
For s(x) = 4:
x = −2.
x = 2.
For s(x) = −1: no real solution.
Check (−2)2 = 22 = 4. Every real square is zero or positive, which independently rules out output −1.
Rung 5Rung 5: solve on a shifted square graph
Use the graph of u(x) = (x − 3)2. Evaluate u(4), then solve u(x) = 4 and u(x) = 0.
The bowl is centered at input 3. Height 4 meets it at inputs 1 and 5, which are not opposites.
What it asks. Read one given input and two given heights. The solution inputs need not be opposites.
Plan. Use vertical lookup for u(4), then horizontal lookup for output heights 4 and 0.
The graph’s bottom is at x = 3, rather than x = 0.
At height 4, the contacts lie two units to either side of 3.
At input 4, read the point (4, 1), so u(4) = 1.Evaluation uses the point’s height as its output.
Follow height 4 to (1, 4) and (5, 4), so x = 1 or x = 5.Solving collects every horizontal input address at the requested height.
At height 0, the graph has only the point (3, 0), so x = 3.The lowest point gives one contact at this height.
Answer
u(4) = 1.
For u(x) = 4:
x = 1.
x = 5.
For u(x) = 0: x = 3.
Check Use the formula to check: (4 − 3)2 = 1, (1 − 3)2 = 4, (5 − 3)2 = 4, and (3 − 3)2 = 0.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: g(3) = 6 because 6 appears somewhere in the table.
A value elsewhere in the output row need not belong to input 3.
✓ Instead: Stay in the column under 3: g(3) = 7.
✗ Not this: A graph solution must always exist.
Some requested heights never meet the graph.
✓ Instead: For s(x) = x2, output −1 has no real input.
✗ Not this: For u(x) = (x − 3)2, solving u(x) = 4 gives x = −2 or x = 2.
The graph is centered at input 3. Those two proposed inputs give outputs 25 and 1.
✓ Instead: Height 4 meets the graph at x = 1 and x = 5: (1 − 3)2 = (5 − 3)2 = 4.
Tips and tricks
Remember down to evaluate, up from every match to solve in an input row above an output row.
On a graph, evaluation fixes horizontal position; solving fixes height.
Trap. Reading only the first matching output when solving. Scan the whole table or the whole horizontal level of the graph before writing the solution set.
