Elementary Functions Part 1, Functions Lecture 1.1c, Finding the domains of functions Dr. Ken W. Smith Sam Houston State University 2013 Smith (SHSU) Elementary Functions 2013 22 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Domains of functions The domain of a function is (generally) the largest possible set of inputs into the function. Let’s find the domain of the function f ( x ) = √ x. It is often easier to ask the question, “What is not in the domain?”. For the function f ( x ) = √ x we ask the question, “Which real numbers do not have a square root?” We cannot evaluate f ( x ) at negative numbers since the square of a real number cannot be negative. So the domain must be numbers which are not negative, that is, zero and positive real numbers. (We can indeed take the square root of 0 so we want to include 0 in the domain.) We can write our answer in interval notation: Solution. The domain of f ( x ) = √ x is [0 , ∞ ) . Smith (SHSU) Elementary Functions 2013 23 / 27
Definition of a function x + 2 + 2 x − 3 1 Example. Find the domain of the function g ( x ) = 2 x + 1 + x − 5 . Solution. What numbers cannot serve as input to g ( x )? Since we cannot have denominators equal to zero then x = − 2 cannot be an input; neither can x = − 1 2 . So the domain of this function g is all real numbers except x = − 2 and x = − 1 2 . There are several ways to write the domain of g . Using set notation, we could write the domain as { x ∈ R : x � = − 2 , − 1 2 } . This is a precise symbolic way to say, “All real numbers except − 2 and − 1 2 . ” We could also write the domain in interval notation: ( −∞ , − 2) ∪ ( − 2 , − 1 2) ∪ ( − 1 2 , ∞ ) . This notation says that the domain includes all the real numbers smaller than − 2 , along with all the real numbers between − 2 and − 1 2 , along with (in addition) the real numbers larger than − 1 2 . Smith (SHSU) Elementary Functions 2013 24 / 27
Definition of a function x + 2 + 2 x − 3 1 Example. Find the domain of the function g ( x ) = 2 x + 1 + x − 5 . Solution. What numbers cannot serve as input to g ( x )? Since we cannot have denominators equal to zero then x = − 2 cannot be an input; neither can x = − 1 2 . So the domain of this function g is all real numbers except x = − 2 and x = − 1 2 . There are several ways to write the domain of g . Using set notation, we could write the domain as { x ∈ R : x � = − 2 , − 1 2 } . This is a precise symbolic way to say, “All real numbers except − 2 and − 1 2 . ” We could also write the domain in interval notation: ( −∞ , − 2) ∪ ( − 2 , − 1 2) ∪ ( − 1 2 , ∞ ) . This notation says that the domain includes all the real numbers smaller than − 2 , along with all the real numbers between − 2 and − 1 2 , along with (in addition) the real numbers larger than − 1 2 . Smith (SHSU) Elementary Functions 2013 24 / 27
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