What is the difference between an implicit and explicit domain in a function?

In summary, Ron Larson explains that a function's domain can either be explicitly defined or implied by the expression used to define the function. The implied domain is the set of all real numbers for which the expression is defined. Explicitly defining a function's domain means stating it explicitly, while an implicit definition is implied by the expression used. This difference can be seen in examples of functions with implicit or explicit domains.
  • #1
nycmathguy
Homework Statement
What is the difference between defining a function explicitly and implicitly?
Relevant Equations
n/a
Ron Larson stated:

"The domain of a function can be described explicitly or it can be implied by the
expression used to define the function. The implied domain is the set of all real
numbers for which the expression is defined."

1. How is a function defined explicitly?

2. How is a function defined implicitly?

3. What's the basic difference between the two definitions?
 
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  • #2
nycmathguy said:
Homework Statement:: What is the difference between defining a function explicitly and implicitly?
Relevant Equations:: n/a

Ron Larson stated:

"The domain of a function can be described explicitly or it can be implied by the
expression used to define the function. The implied domain is the set of all real
numbers for which the expression is defined."
Emphasis added above. Also, your homework statement above doesn't match the Larson quote.
nycmathguy said:
1. How is a function defined explicitly?
Larson is talking about the domain of a function being defined implicitly or explicitly.
nycmathguy said:
2. How is a function defined implicitly?
See above.
nycmathguy said:
3. What's the basic difference between the two definitions?
Does Larson give any examples of functions being defined with implicit domains or explicit domains?
 

FAQ: What is the difference between an implicit and explicit domain in a function?

1. What is an implicit domain in a function?

An implicit domain in a function is the set of all input values that can be used in the function without causing an error or undefined result. It is not explicitly defined, but rather implied by the function's equation or rule.

2. What is an explicit domain in a function?

An explicit domain in a function is the set of all input values that are explicitly stated or defined to be used in the function. It is typically listed as the domain of the function in its definition.

3. How do implicit and explicit domains differ?

The main difference between implicit and explicit domains is that implicit domains are not explicitly stated or defined, while explicit domains are specifically listed as the set of input values for a function.

4. Can a function have both an implicit and explicit domain?

Yes, a function can have both an implicit and explicit domain. The implicit domain may be larger than the explicit domain, as the function may have additional input values that are not explicitly stated but still result in a defined output.

5. Why is it important to consider the domain of a function?

The domain of a function is important because it determines the set of input values that can be used in the function to produce a meaningful output. It also helps to identify any potential errors or undefined results that may occur when using the function.

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