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Understanding Function Equality in Mathematicse

Function equality is a key concept in mathematics that determines when two functions are considered exactly the same. Two functions are equal only if they share the same domain and produce identical outputs for every input in that domain. This article explains the definition of function equality, how to compare functions, and provides examples using a clean HTML structure.

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~2 min read • Updated Feb 9, 2026

1. What Is Function Equality?


Function equality means that two functions behave exactly the same. Formally, two functions f and g are equal if both of the following conditions are true:


  • They have the same domain
  • For every input x in the domain, their outputs are equal: f(x) = g(x)

2. Why Domain Matters


Even if two functions produce the same formula, they are not equal unless their domains match. The domain defines where the function is valid, so differences in domain create differences in the function itself.


Example:

f(x) = √x   with domain x ≥ 0
g(x) = √x   with domain x ≥ 4

Although the formulas look identical, f and g are not equal because their domains differ.


3. Equality Through Simplification


Sometimes two functions appear different but are actually equal after simplification. If their simplified forms match and their domains are the same, the functions are equal.


Example:

f(x) = (x^2 - 1) / (x - 1)
g(x) = x + 1

At first glance, these look different. But simplifying f(x) gives:


(x^2 - 1) = (x - 1)(x + 1)
f(x) = (x - 1)(x + 1) / (x - 1) = x + 1

However, note that f(x) is undefined at x = 1 because of the denominator. So the domains differ:


  • f(x) domain: all real numbers except 1
  • g(x) domain: all real numbers

Therefore, f and g are not equal even though their simplified formulas match.


4. How to Check If Two Functions Are Equal


  • Compare their domains
  • Simplify both functions if possible
  • Check whether f(x) = g(x) for every value in the domain

5. Example of Equal Functions


Consider the following functions:

f(x) = 2x + 4
g(x) = 2(x + 2)

After simplification:

g(x) = 2x + 4

Both functions have the same formula and the same domain (all real numbers), so:


Conclusion: f(x) and g(x) are equal functions.


6. Conclusion


Function equality requires more than matching formulas. Two functions are equal only when they share the same domain and produce identical outputs for every input. Understanding this concept is essential for algebra, calculus, and advanced mathematical analysis.


Written & researched by Dr. Shahin Siami

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