I was solving exercises of pre calculus book when I came up with a question which asked me to find the implied domain. Please can anyone tell me? What's the exact difference between the two? I think they are same.
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$\begingroup$The implied domain is the largest possible subset of the real numbers where each member of the subset yields a real number when the function is applied to it. The exact definition depends on how the function is defined, such as a formula, verbal description, or other.
The domain of a function is the set of the first values in the ordered pairs that make the function. This definition applies to the "univocal set of ordered pairs" concept of a function.
The difference usually occurs when some limitation is placed on a domain. Examples are saying time or length is positive, the number of people must be integral and less than the current population of the earth. Such a function with limitations on the domain is often called a restricted function.
If a formula is given without any limitation/restriction given in the problem or by common sense, we use the implied domain.
$\endgroup$ $\begingroup$The implied domain of a function is the set of all real numbers which satisfy the expression. For example, $f(x) = \frac{x^2 - 4}{x-2}$ has an implied domain of all reals except 2. However, we can simplify $f(x)$ to just $x + 2$ by eliminating $x-2$ from the top and bottom. When we do this, we have to add the exclusion that x cannot equal 2. So the domain for $f(x)$ would be explicitly defined if it looked like this: $f(x) = x + 2, x \neq 2$.
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