Why Compare Different Representations?
A function can be presented by a formula, a table of input-output pairs, or a graph. Each representation highlights different information. A formula gives a rule for calculating outputs, a table records particular values, and a graph displays the function’s input-output pairs visually.
When representations are intended to describe the same function, they should agree wherever they make a claim. A point \((a,b)\) in a table should correspond to the formula value \(f(a)=b\), and the graph should contain that point. But a short table usually contains only a few inputs. Agreement at those inputs is useful evidence of a match; it does not by itself show that the formula and graph agree at every input.
What Does It Mean for Representations to Match?
A function \(f:D_f\to\mathbb{R}\) has domain \(D_f\), the set of inputs for which it is defined. Its graph is the set of ordered pairs formed from every input in that domain and its corresponding output:
Thus, a graph represents a function by displaying its input-output pairs. A point \((a,b)\) belongs to the graph exactly when \(a\in D_f\) and \(f(a)=b\). A table containing some of these pairs records selected points from the graph, unless it explicitly specifies that the listed inputs are the entire domain.
This criterion explains why a few matching values are not enough to establish that two formulas define the same function on all real numbers. They might agree at the listed inputs and differ elsewhere. Conversely, if a graph is explicitly identified as the complete graph of a function and its domain and outputs match the formula at every input, then the representations agree.
A Practical Matching Method
In many exercises, you are given a formula, a short table, and a graph or a verbal description of a graph. Use checks that move from individual points to broader features:
Find which inputs the formula allows, which inputs the table lists, and whether the graph shows endpoints, gaps, or a restricted interval.
Substitute the table’s inputs into the formula. Each listed output should equal the formula value, and the corresponding graph point should have the same coordinates.
Check whether the graph’s shape and direction are plausible for the formula. A few points can fit several different curves, so use the full visible pattern and any domain features.
Say whether the checks support a match at the listed inputs or establish agreement over the full domain. Do not treat a finite table as a complete description unless its domain is specified to be finite.
Worked Example: Matching a Linear Formula, Table, and Graph
Worked Example: Matching a Linear Formula, Table, and Graph
Compare the formula \(f(x)=2x-1\), the table below, and a graph described as a straight line passing through \((-2,-5)\), \((0,-1)\), and \((3,5)\), extending in both directions.
| \(x\) | Table output |
|---|---|
| \(-2\) | \(-5\) |
| \(0\) | \(-1\) |
| \(3\) | \(5\) |
First calculate the formula’s output at each listed input:
Each result agrees with the table. The graph contains the corresponding ordered pairs, so the formula, table, and graph agree at these three inputs.
To check that the graph’s stated line has the right rule, use two of its points. The slope between \((-2,-5)\) and \((0,-1)\) is
The line also passes through \((0,-1)\), so its vertical intercept is \(-1\). A line with slope \(2\) and vertical intercept \(-1\) has equation \(y=2x-1\), which is the given formula. As a check, substituting \(x=3\) gives \(y=2(3)-1=5\), the graph’s stated third point.
The formula and described line have domain \(\mathbb{R}\). The table lists only \(-2,0,3\), so it records selected values rather than the full domain. The formula reproduces every listed table value, and the described line agrees with the formula. The finite table supports the match at its listed inputs but, by itself, does not establish agreement at every real input.
Domain and Shape Can Rule Out a Match
Matching values is only one part of the comparison. Domain features are especially important when a formula excludes inputs or a graph has an endpoint or a gap. For example, if a formula is defined only for \(x\geq 1\), a graph that continues to the left of \(x=1\) cannot be its complete graph. Likewise, if the formula is defined at \(x=1\), the graph must include a point with that input.
Overall shape is another useful check. A straight-line pattern, a curved pattern, and a graph that changes rule at a boundary have different visual features. However, appearance is not a substitute for checking coordinates: several functions can pass through the same finite collection of points.
Worked Example: Matching a Piecewise Rule and Its Boundary
Worked Example: Matching a Piecewise Rule and Its Boundary
Consider the function
Compare it with a table listing \(x=-1,0,1,2\), with outputs \(1,0,1,3\), and a graph described as following the parabola \(y=x^2\) for \(x\leq1\) and the line \(y=2x-1\) for \(x>1\).
Evaluate the formula using the branch that applies to each input:
These values reproduce all four table entries. The graph description also uses the same rules on the same intervals. At the boundary \(x=1\), the first branch applies, so the function includes the point \((1,1)\). The second branch is used only for \(x>1\); its boundary point at \(x=1\) is not part of that branch. Its formula would give \(2(1)-1=1\), the same height, but the domain condition still determines which branch includes the boundary.
Both branches together cover every real input: inputs at most \(1\) use the first rule, and inputs greater than \(1\) use the second. Therefore the function’s domain is \(\mathbb{R}\). The table lists only four inputs, so it does not describe that entire domain. Its values agree with the formula, and the described graph has the matching branches and boundary convention. These checks support the match without claiming that four table entries alone establish the full graph.
Worked Example: Checking a Restricted Domain and Curved Pattern
Worked Example: Checking a Restricted Domain and Curved Pattern
Compare \(h(x)=\sqrt{x-1}\) with a table listing \((1,0)\), \((2,1)\), and \((5,2)\), and a graph described as starting at \((1,0)\), extending to the right, and increasing with a curved shape.
The square root requires its input to be nonnegative. Thus \(x-1\geq0\), which gives \(x\geq1\). The formula’s domain is \([1,\infty)\). Now evaluate it at the table inputs:
All three outputs agree with the table. The graph’s starting point \((1,0)\) matches the formula at its smallest permitted input. Its extension to the right is consistent with the domain \(x\geq1\), and its increasing curved pattern is consistent with the square-root rule. These are useful checks of both values and domain.
The formula is defined for every real \(x\geq1\), while the table records only three inputs. The graph description suggests the same domain and shape, but a verbal summary and a finite set of points do not specify every point of a graph. The checks support the match; they should not be mistaken for a proof based only on the table.
A Short Justification for the Matching Criterion
The definition of a function explains why domain and outputs are both required. A function consists of a specified set of inputs together with one assigned output for each input. If two functions have the same domain and assign the same output to every input in that domain, then they contain exactly the same ordered pairs in their graphs.
For the reverse direction, if the graphs are the same sets of ordered pairs, then they contain the same first coordinates, so the domains are equal. For any input \(x\) in that domain, the graph contains the pair \((x,f(x))\). Since the graphs are equal, it also contains the pair assigned by \(g\). A function assigns only one output to each input, so \(f(x)=g(x)\). Therefore equal graphs and equal functions amount to the same domain and the same output at every input.
A table with just a few rows checks this condition only at its listed inputs. It can confirm those values, but it cannot verify the condition at unlisted inputs unless the table gives the entire domain.
Common Pitfall: Treating Listed Inputs as the Whole Domain
Suppose a table for \(f(x)=2x-1\) lists \(x=-2,0,2\), with outputs \(-5,-1,3\). It is incorrect to conclude, “The formula and table have the same domain, so they are the same function,” because the formula’s domain is \(\mathbb{R}\), whereas the table lists only three inputs.
The correction is to say that the formula matches every value listed in the table. The formula has domain \(\mathbb{R}\); the table provides selected input-output pairs and does not establish the function’s values at all other real inputs. If the table is explicitly stated to be the complete function table, then its domain is the three listed inputs, and it represents a function with that finite domain—not the formula \(2x-1\) on all real numbers.
Why This Skill Matters
Being able to move among formulas, tables, and graphs helps you check whether different descriptions tell a consistent story. A table can make a calculation transparent; a graph can reveal endpoints and broad patterns; a formula can make the rule and its domain precise. Using all three views can uncover a mismatch that might be overlooked in any one representation.
The most careful conclusion is also the most informative. State exactly what you checked: for example, “The formula matches all listed table values, and the graph has the stated domain and pattern.” Reserve the claim that two functions are equal for cases where their domains and outputs agree throughout the domain.
Check Your Understanding
For each question, distinguish checks at selected inputs from claims about the full function.
- If a formula and table agree at four inputs, what has been verified, and what has not necessarily been verified?
- What two conditions must hold for two functions to be the same function?
- For \(q(x)=\sqrt{x+3}\), what domain feature should you look for in a matching graph?
- In a piecewise rule, why should you check which branch includes a boundary input?
- A graph extends to inputs less than \(0\), while the formula is defined only for \(x\geq0\). Can the graph be the complete graph of that formula? Explain.
