Why Input Choice Matters
A table can help reveal what a function does near a target, but the table is only as useful as the inputs it contains. Inputs that are far from the target may not describe nearby behavior well. Inputs from only one side may miss a different pattern on the other side.
To probe a target \(a\), choose inputs both less than and greater than \(a\), with distances from \(a\) that become progressively smaller. Then calculate the corresponding function values. The aim is to observe how the outputs behave as the inputs get close—not to substitute \(a\) into the function.
Constructing Inputs on Both Sides
A convenient way to make paired inputs is to choose a positive distance \(h\) and use \(a-h\) on the left and \(a+h\) on the right. For example, distances \(h=0.1,\ 0.01,\ 0.001\) generate three pairs of inputs. The smaller \(h\) becomes, the closer each member of the pair is to \(a\).
The distance check is exact: \(|(a-h)-a|=h\) and \(|(a+h)-a|=h\). Since \(h\) is positive and decreases, these inputs get closer to \(a\) from their respective sides. This construction gives a systematic way to choose test inputs rather than selecting decimals at random.
Paired inputs do not have to be perfectly symmetric. If the target is \(1.5\), for instance, the inputs \(1.49\) and \(1.501\) are both useful, even though their distances are different. What matters is knowing which side each input is on and how far away it is.
Worked Example: Probing \(x^2\) Near \(1.5\)
Worked Example: Probing \(x^2\) Near \(1.5\)
Choose several inputs on both sides of \(1.5\), moving progressively closer, and calculate the corresponding values of \(f(x)=x^2\).
The distances are \(0.1,\ 0.01,\ 0.001\), with an additional pair \(0.1\) away to show how a wider probe compares. For the closest pair, \(1.5-0.001=1.499\) and \(1.5+0.001=1.501\).
| Input \(x\) | Side of \(1.5\) | Distance \(|x-1.5|\) | Calculation of \(f(x)=x^2\) | Function value |
|---|---|---|---|---|
| 1.4 | Left | 0.1 | \(1.4^2=1.96\) | 1.96 |
| 1.49 | Left | 0.01 | \(1.49^2=2.2201\) | 2.2201 |
| 1.499 | Left | 0.001 | \(1.499^2=2.247001\) | 2.247001 |
| 1.501 | Right | 0.001 | \(1.501^2=2.253001\) | 2.253001 |
| 1.51 | Right | 0.01 | \(1.51^2=2.2801\) | 2.2801 |
| 1.6 | Right | 0.1 | \(1.6^2=2.56\) | 2.56 |
The closest pair produces values \(2.247001\) and \(2.253001\), which are closer to each other than the values from the wider pair. The table suggests that the function values on both sides are approaching \(2.25\). The target input \(1.5\) was not needed to make this observation.
The arithmetic can be checked directly: \(1.499^2=(1.5-0.001)^2=2.25-0.003+0.000001=2.247001\), and \(1.501^2=(1.5+0.001)^2=2.25+0.003+0.000001=2.253001\).
Why the Inputs Reveal Nearby Behavior
The construction of the inputs explains why this probing method is meaningful. Write an input near \(1.5\) as \(x=1.5+h\), where \(h\) is the signed distance from \(1.5\). A negative \(h\) gives an input to the left; a positive \(h\) gives an input to the right. For the function in the example, the difference between the output and \(2.25\) is
This identity matches the example’s target and function. It also gives a precise bound on how far the output can be from \(2.25\). If \(|h|<1\), then \(-1<h<1\), so \(2<3+h<4\) and therefore \(|3+h|<4\). It follows that
Thus, as the input distance \(|h|\) decreases, this bound forces the output’s distance from \(2.25\) to decrease as well. More explicitly, given any desired positive output tolerance \(\varepsilon\), choose \(\delta=\min(1,\varepsilon/4)\). If \(0<|x-1.5|<\delta\), then \(h=x-1.5\), so \(|h|<1\) and \(|h|<\varepsilon/4\). Consequently,
This derivation explains why progressively closer inputs are useful for this function: a sufficiently small input distance guarantees an arbitrarily small output distance from \(2.25\). A table provides sample evidence; the calculation explains the pattern for every input close enough to the target.
Worked Example: Probing a Function Undefined at the Target
Worked Example: Probing a Function Undefined at the Target
Consider \(g(x)=\dfrac{x^2-1}{x-1}\). The expression is undefined at \(x=1\), so choose inputs on both sides and do not evaluate \(g(1)\).
Use distances \(0.1,\ 0.01,\ 0.001\). For each input, calculate the numerator and denominator separately before dividing.
| Input \(x\) | Numerator \(x^2-1\) | Denominator \(x-1\) | Quotient \(g(x)\) |
|---|---|---|---|
| 0.9 | \(0.9^2-1=0.81-1=-0.19\) | \(0.9-1=-0.1\) | \((-0.19)/(-0.1)=1.9\) |
| 0.99 | \(0.99^2-1=0.9801-1=-0.0199\) | \(0.99-1=-0.01\) | \((-0.0199)/(-0.01)=1.99\) |
| 0.999 | \(0.999^2-1=0.998001-1=-0.001999\) | \(0.999-1=-0.001\) | \((-0.001999)/(-0.001)=1.999\) |
| 1.001 | \(1.001^2-1=1.002001-1=0.002001\) | \(1.001-1=0.001\) | \(0.002001/0.001=2.001\) |
| 1.01 | \(1.01^2-1=1.0201-1=0.0201\) | \(1.01-1=0.01\) | \(0.0201/0.01=2.01\) |
| 1.1 | \(1.1^2-1=1.21-1=0.21\) | \(1.1-1=0.1\) | \(0.21/0.1=2.1\) |
The closest probes give \(1.999\) from the left and \(2.001\) from the right. These values suggest that the outputs on both sides are near \(2\). The input \(x=1\) remains excluded: the original expression would have denominator \(1-1=0\), so it does not define a function value there.
As a check on each quotient, multiplying the reported value by its denominator recovers its numerator. For example, \(1.999(-0.001)=-0.001999\), and \(2.001(0.001)=0.002001\). The other rows check in the same way: \(1.9(-0.1)=-0.19\), \(1.99(-0.01)=-0.0199\), \(2.01(0.01)=0.0201\), and \(2.1(0.1)=0.21\).
Use Both Sides to Detect Different Patterns
Testing from the left and right is useful not only when the outputs seem to move toward a shared value. Sometimes the two sides show different behavior. Such evidence is a reason to examine each side separately and choose additional inputs; a few probes alone should not be treated as a complete argument about every nearby input.
Worked Example: A Different Output on Each Side
For \(x\ne1\), let \(q(x)=\dfrac{x-1}{|x-1|}\). Probe the target \(1\) using distances \(0.2,\ 0.02,\ 0.002\) on both sides.
| Input \(x\) | Distance from 1 | Calculation | Function value |
|---|---|---|---|
| 0.8 | 0.2 | \((-0.2)/|{-0.2}|=(-0.2)/0.2\) | -1 |
| 0.98 | 0.02 | \((-0.02)/|{-0.02}|=(-0.02)/0.02\) | -1 |
| 0.998 | 0.002 | \((-0.002)/|{-0.002}|=(-0.002)/0.002\) | -1 |
| 1.002 | 0.002 | \(0.002/|0.002|=0.002/0.002\) | 1 |
| 1.02 | 0.02 | \(0.02/|0.02|=0.02/0.02\) | 1 |
| 1.2 | 0.2 | \(0.2/|0.2|=0.2/0.2\) | 1 |
Every listed left-hand input produces \(-1\), while every listed right-hand input produces \(1\). The output does not get closer to one common value in these samples. This example shows why it is important to probe both sides rather than letting one side stand in for the other. The target itself is excluded because \(q(1)\) would require division by zero.
Choosing the Number of Decimal Places
More decimal places do not automatically make a table more informative. The purpose of adding digits is to reduce the distance from the target in a controlled way. A sequence such as \(a-0.1,\ a-0.01,\ a-0.001\) on the left and the corresponding plus inputs on the right makes that progression visible.
For practical calculations, keep enough precision in the inputs and outputs to distinguish the observed pattern. Do not replace the input by the target just because the decimal is close. Also record the actual input distance: \(1.999\) is \(0.001\) from \(2\), whereas \(1.99\) is \(0.01\) from \(2\). That distinction helps you judge which output values come from the closer probes.
When a function is undefined at the target, first check that each chosen input is in the function’s domain. A point close to the target may still produce an undefined value if it is another excluded input. A table can only use inputs at which the function is defined.
A Common Mistake: Substituting the Target
A frequent error is to evaluate at the target even when the task is to study nearby behavior. For the function \(g(x)=\dfrac{x^2-1}{x-1}\), someone might write
The calculation correctly shows that \(g(1)\) is undefined, but the conclusion is wrong: an undefined target value does not prevent us from probing the function at nearby inputs. Use \(0.99\) and \(1.01\), for example. Their distances from \(1\) are \(0.01\), and the table above gives \(g(0.99)=1.99\) and \(g(1.01)=2.01\). These values provide information about behavior near the target without assigning a value at the target.
A Practical Input-Selection Routine
Identify the number \(a\) whose nearby behavior you want to investigate.
Use positive distances that decrease, such as \(0.1,\ 0.01,\ 0.001\).
For each distance \(h\), try \(a-h\) and \(a+h\), provided the function is defined there.
Keep the input, its distance from the target, and the function value in separate columns.
Check whether values from each side become closer to a common number, remain separated, or show another pattern. Treat the table as evidence from selected inputs, not as a substitute for further reasoning.
Check Your Understanding
For each question, focus on the choice of inputs and what the resulting table can show.
- For a target \(a=4\), write three input pairs on opposite sides whose distances from \(4\) decrease.
- What are the distances from \(2\) for the inputs \(1.9,\ 1.99,\ 2.001\)? Which input is closest?
- If a function is undefined at \(x=1\), why can inputs such as \(0.99\) and \(1.01\) still be useful for probing behavior near \(1\)? What should be checked before calculating their outputs?
- Suppose a table lists inputs only to the left of a target. What useful information is missing?
- For \(f(x)=x^2\) near \(1.5\), write \(x=1.5+h\) and expand \(x^2-2.25\). What does the resulting expression indicate about the output’s distance from \(2.25\) when \(|h|<1\)?
