How can I tell whether two function formulas define the same function?
Quick answer
Two function formulas define the same function only if they have the same domain and give the same output for every input in that domain. Checking a few sample values is not enough. First identify the allowed inputs for each formula; then compare their outputs across that shared domain. For example, cancelling a factor can make formulas look identical while one still excludes an input.
The question, the details, the next step.
What should I check when I want to know whether two formulas define the same function? I’ve tried plugging in a few numbers, but that seems inconclusive.
Use a two-part test: the functions must have the same domain, and their outputs must match for every input in that domain. A function assigns one output to each input in its domain, so matching a handful of input-output pairs cannot establish equality. [S1]
Start by writing down the allowed inputs for each formula. Then compare the rules on those inputs. If the formulas have different domains, they do not define the same function, even if their outputs match wherever both are defined. A simplified-looking formula still needs a domain check as well as an algebra check.
Can you show how that works with formulas that look algebraically equivalent?
Assume both formulas are intended as real-valued functions of a real input. Compare:
f(x) = x^2 - 1
and
g(x) = (x^2 - 1)/(x - 1).
For f, every real input is allowed. For g, the denominator cannot be zero, so x = 1 is excluded. Factoring gives x^2 - 1 = (x - 1)(x + 1), and cancellation shows g(x) = x + 1 for inputs other than 1. At those allowed inputs, f also equals x + 1. But at x = 1, f(1) = 0 while g(1) is undefined. So the formulas agree wherever g is defined, but the functions are not the same because their domains differ. [S1][S2]
If I test some values and simplify both formulas to the same expression, is that ever enough?
Not by itself. Sample values can help you spot an error, and simplifying can help show that outputs match, but neither step replaces checking the domains and outputs for every allowed input. For the example, testing x = 0 gives -1 from both formulas, but that match does not reveal that g excludes 1.
A useful proof pattern is: first state each domain; if they differ, the functions are not equal. If they match, show that the outputs agree for every input in that common domain. In this example, cancellation justifies g(x) = x + 1 only when x ≠ 1; it does not restore the excluded input. [S1][S2]
Can you give me a checklist I can use on homework, including what to write if the domains differ?
Try this reusable comparison:
- State the setting: for example, “Assume real inputs.”
- Find each domain: note restrictions such as a zero denominator or a negative radicand under an even root. [S2]
- Compare the domains: if they differ, say the functions are not equal, even if their rules agree on the overlap.
- If the domains match, compare outputs: show the formulas give equal values for every input in that domain, not just for test values.
- Keep restrictions after simplification: cancellation does not add a previously excluded input back to the domain.
For the worked example, a concise conclusion is: “The domains differ: f allows x = 1, but g does not. Therefore, these formulas do not define the same function, although they agree for every input where g is defined.”
A practical checklist
State the assumed input set, such as the real numbers.
Find and write the domain of each formula separately.
Check whether the domains are identical before comparing rules.
If domains match, prove outputs agree for every allowed input.
Retain excluded inputs after factoring or cancellation.
A few more useful answers
Do two formulas have to look the same to define the same function?
No. Different-looking formulas can give the same outputs on the same domain. Check the domains first, then establish that the outputs match for every input there.
Can a graph or a few table entries prove two functions are equal?
A graph or selected entries can help compare examples, but they do not by themselves establish equality for every input. Use them as checks, not as a replacement for comparing the domains and outputs.
Sources
References used for this guide.
- [S1] OpenStax Precalculus 2e: Functions and Function Notation openstax.org · Checked
- [S2] OpenStax Precalculus 2e: Domain and Range openstax.org · Checked