Can we be functionally equivalent?

📌 CONCEPT: Two functions are considered functionally equivalent if they have the same domain and produce the same output for every input, even though their equations may look different.

📐 RULE / FORMULA: For two functions f(x) and g(x) to be functionally equivalent, it must be true that f(x) = g(x) for all real numbers in their common domain.

💡 WORKED EXAMPLE: To check if f(x) = 3x + 2 and g(x) = 2x + 4 are functionally equivalent, we need to see if f(x) = g(x) for all real numbers. By equating their equations and verifying the equality, we can confirm that they produce the same output for every input, making them functionally equivalent.

⚠️ COMMON MISTAKE: Students often mistakenly assume that functionally equivalent functions must be identical in their equations, when in fact they can have different forms as long as their outputs are the same for every input.