Reading the Highest and Lowest Attained Values
A graph can make a function’s overall behavior visible, and a table can organize its values at particular inputs. In either case, reading absolute extrema requires more than spotting a high or low-looking point. You must know the function’s domain, decide whether a candidate value is actually attained, and distinguish the output value from the input where it occurs.
This distinction matters at open endpoints and in tables that show only selected inputs. A function may approach a value without ever taking it, or a table may omit values between its listed rows. In those situations, a visible bound or a largest listed value is not automatically an absolute extremum of the function.
The phrase “for every \(x\in D\)” is essential: an absolute extremum is greatest or least across the entire domain, not just in a nearby part of the graph or among a few selected table entries. A function can have more than one location where the same absolute extreme occurs. It can also have no absolute maximum, no absolute minimum, or neither.
Value and Location Are Different
If \(f(2)=7\) and \(7\) is the greatest output on the domain, then \(7\) is the absolute maximum value, while \(x=2\) is a location where it occurs. When two inputs both give the greatest output, the maximum value is still one number, but there are two locations.
On a graph, the vertical coordinate gives the function value and the horizontal coordinate gives its input. Thus, the highest included point or section of the graph determines the maximum value and its location or locations. The same reading applies at the bottom of the graph for a minimum.
Worked Example: Reading Extrema from a Piecewise Linear Graph
Worked Example: Reading Extrema from a Piecewise Linear Graph
Suppose a graph consists of line segments joining the filled points \((-2,1)\), \((-1,4)\), \((1,4)\), \((2,-2)\), and \((3,4)\), in that order. Find its absolute maximum and minimum values and all their locations.
Step 1: Compare the heights at the joined points. The listed output values, in order, are \(1,4,4,-2,4\). The greatest listed height is \(4\); the least is \(-2\).
Step 2: Check the segments between the points. On a line segment, the output values lie between the outputs at its endpoints. Therefore, none of the segments joining these points can rise above the greatest endpoint height \(4\) or fall below the least endpoint height \(-2\).
Step 3: Find every location attaining each value. The segment from \((-1,4)\) to \((1,4)\) is horizontal, so \(f(x)=4\) for every \(x\in[-1,1]\). The filled point \((3,4)\) also attains \(4\). The value \(-2\) occurs at the filled point \((2,-2)\).
Answer: The absolute maximum value is \(4\), attained for \(x\in[-1,1]\) and at \(x=3\). The absolute minimum value is \(-2\), attained at \(x=2\).
Check: Every segment’s endpoint heights are between \(-2\) and \(4\), and the graph includes points at both of those extreme heights. Thus, the reported values are attained and no segment exceeds them.
Open and Filled Points: Is the Value Attained?
A filled point indicates that the function includes that input-output pair. An open circle indicates that the graph approaches that point but does not include it. An open circle can show an important bound, but its height is not a function value at that input.
For example, if a graph approaches height \(5\) near an open endpoint but all its actual heights are below \(5\), then \(5\) is not an absolute maximum value. It may be an upper bound, and it may be the least upper bound, but no input produces the value \(5\).
When the greatest output exists, it is also the supremum and is attained. When the least output exists, it is also the infimum and is attained. If the graph only approaches a bound, do not call that bound an absolute extreme value.
Worked Example: A Graph with Unattained Bounds
Worked Example: A Graph with Unattained Bounds
Suppose the graph is the line \(y=x\) for \(0<x<4\), with open circles at \((0,0)\) and \((4,4)\). Determine whether it has an absolute maximum or minimum, and identify its bounds.
Step 1: Read the domain and outputs. The allowed inputs satisfy \(0<x<4\). Since \(f(x)=x\), the outputs also satisfy \(0<f(x)<4\).
Step 2: Test the apparent top and bottom values for attainment. The graph approaches \(4\) as \(x\) approaches \(4\), but \(x=4\) is excluded, so no input produces \(f(x)=4\). It approaches \(0\) as \(x\) approaches \(0\), but \(x=0\) is excluded, so no input produces \(f(x)=0\).
Step 3: Decide whether a greatest or least output exists. For any input \(x\) in \((0,4)\), the input \(x'=(x+4)/2\) is also in \((0,4)\) and satisfies \(x<x'<4\). Hence \(f(x')>f(x)\), so no output is greatest. Likewise, \(x''=x/2\) is in \((0,4)\) and satisfies \(0<x''<x\), so \(f(x'')<f(x)\); no output is least.
Answer: The function has no absolute maximum and no absolute minimum. Its outputs have supremum \(4\) and infimum \(0\), but neither bound is attained.
Check: The range is \((0,4)\). Neither endpoint belongs to that interval, while every range value is between them, confirming that \(0\) and \(4\) are the infimum and supremum but not extrema.
Reading Absolute Extrema from Tables
A table provides values at the inputs it lists. First identify what kind of domain the table represents. If its rows give the complete domain, compare all listed outputs: the greatest output is an absolute maximum and the least is an absolute minimum. If rows are only selected measurements or sample points from a larger domain, the table alone cannot establish the function’s extrema between those inputs.
Repeated greatest or least outputs mean that the corresponding extreme occurs at every listed input producing that output. Report all such locations when the table gives a complete domain. If the table is only a sample, describe the largest or smallest listed value rather than claiming it is a function-wide extreme.
Worked Example: Finding Extrema from a Complete Table
A function is defined only on the inputs in this table. Find its absolute extrema and their locations.
| \(x\) | \(-2\) | \(0\) | \(1\) | \(4\) | \(6\) |
|---|---|---|---|---|---|
| \(f(x)\) | \(3\) | \(-1\) | \(5\) | \(5\) | \(2\) |
Step 1: Confirm the domain. The table specifies that the function’s complete domain is \(\{-2,0,1,4,6\}\), so there are no unlisted inputs whose outputs must be considered.
Step 2: Compare the outputs. The outputs are \(3,-1,5,5,2\). Their greatest value is \(5\), and their least value is \(-1\).
Step 3: Find all inputs producing those values. The output \(5\) occurs at \(x=1\) and \(x=4\). The output \(-1\) occurs at \(x=0\).
Answer: The absolute maximum value is \(5\), attained at \(x=1\) and \(x=4\). The absolute minimum value is \(-1\), attained at \(x=0\).
Check: Comparing \(5\) with each other output gives \(5\geq3\), \(5\geq-1\), and \(5\geq2\); comparing \(-1\) with each gives \(-1\leq3\), \(-1\leq5\), and \(-1\leq2\). Thus, both reported values are extreme over the complete domain.
Why the Graph and Table Rules Work
The definitions explain why these reading methods are valid. The graph of \(f\) consists of points \((x,f(x))\). A point with greatest vertical coordinate among all included graph points corresponds to an output at least as large as every other output. If that point is \((c,M)\), then \(f(c)=M\) and \(f(c)\geq f(x)\) for every input \(x\) in the domain. By definition, \(M\) is an absolute maximum and \(c\) is a location where it occurs. Reversing “greatest” to “least” proves the corresponding statement for an absolute minimum.
For a complete finite table, the outputs in its rows are all the function’s outputs. Comparing them therefore checks the defining inequality against every domain input. For a partial table, that reasoning stops at the listed rows: an unlisted input might produce a larger or smaller value. The table by itself cannot rule that out.
A Common Mistake: Counting an Open Endpoint
Worked Example: Correcting an Open-Endpoint Error
Let \(f(x)=x\) on \(0\leq x<1\). A graph has a filled point at \((0,0)\) and an open circle at \((1,1)\). A mistaken reading says, “The absolute maximum is \(1\), because the graph reaches the height \(1\) at its right endpoint.”
What is wrong? The endpoint input \(x=1\) is excluded. The open circle \((1,1)\) is not part of the graph, so \(f(1)=1\) is not a value of this function on its domain.
Correct the claim. For every allowed input \(x<1\), \(f(x)=x<1\), so \(1\) is an upper bound. But no allowed input produces \(1\). Therefore, there is no absolute maximum; the supremum is \(1\). The input \(x=0\) is included and \(f(0)=0\). Also, every allowed input satisfies \(x\geq0\), so \(0\) is the absolute minimum value, attained at \(x=0\).
Check: The range is \([0,1)\). It includes \(0\) and excludes \(1\), confirming the minimum and the unattained upper bound.
A Reliable Reading Routine
Use the following sequence whenever a graph or table is presented. It separates the numerical comparison from the question of whether the candidate is actually included.
Determine which inputs are included. On a graph, distinguish filled points from open circles; in a table, check whether the listed rows are the complete domain or only selected data.
Compare vertical heights on the included graph or output values in a complete table.
Confirm that each candidate is a function value, then list every input where that output occurs.
If the graph approaches a height that it never reaches, report the relevant bound without calling it an absolute maximum or minimum.
Reading extrema accurately is useful well beyond a sketch or data table. It reinforces the distinction between an input and its output, and between a value that is attained and one that is only approached. Those distinctions are essential whenever a problem asks for an absolute extreme on a stated domain.
Check Your Understanding
For each question, pay attention to the domain and whether the candidate value is included.
- A graph has its highest included height at \(y=6\), attained at \(x=-2\) and \(x=5\). State the absolute maximum value and its locations.
- A graph approaches \(y=-3\) at an open endpoint but never goes below \(y=-3\). Does that information alone establish an absolute minimum of \(-3\)? Explain.
- A complete table has outputs \(4,0,4,-2\) at inputs \(1,2,3,4\), respectively. Find the absolute maximum and minimum values and all their locations.
- A table lists measurements at several inputs but does not describe the function between them. Can the largest listed output automatically be called the absolute maximum of the function? Why or why not?
- For \(f(x)=x\) on \(1<x\leq5\), identify its absolute extrema, if they exist, and its unattained bound.
