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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 Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 12 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 12 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 12 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 12 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 12 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 12 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 12 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 21 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 21 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 21 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 21 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 21 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Example. Find the domain of the function g(x) = 1 2x − 3 + + x − 5. x + 2 2x + 1 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 = − 21 . So the domain of this function g is all real numbers except x = −2 and x = − 21 . There are several ways to write the domain of g. Using set notation, we could write the domain as 1 {x ∈ R : x 6= −2, − }. 2 This is a precise symbolic way to say, “All real numbers except −2 and − 21 .” We could also write the domain in interval notation: 1 1 (−∞, −2) ∪ (−2, − ) ∪ (− , ∞). 2 2 This notation says that the domain includes all the real numbers smaller than −2, along with all the real numbers between −2 and − 12 , along with (in addition) the real numbers larger than − 12 . Smith (SHSU) Elementary Functions 2013 24 / 27 Definition of a function Some worked exercises. 1 Find the domain of the function f (x) = √ x−1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞). Smith (SHSU) Elementary Functions 2013 25 / 27 Definition of a function Some worked exercises. 1 Find the domain of the function f (x) = √ x−1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞). Smith (SHSU) Elementary Functions 2013 25 / 27 Definition of a function Some worked exercises. 1 Find the domain of the function f (x) = √ x−1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞). Smith (SHSU) Elementary Functions 2013 25 / 27 Definition of a function Some worked exercises. 1 Find the domain of the function f (x) = √ x−1 Solution. Since the square root function requires nonnegative inputs, we must have x − 1 ≥ 0. Therefore we must have x ≥ 1. The domain is [1, ∞). Smith (SHSU) Elementary Functions 2013 25 / 27 Definition of a function √ 2 Find the domain of the function f (x) = x−1 x−3 Solution. Again, we must have x ≥ 1 but we must also prevent the denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞). Smith (SHSU) Elementary Functions 2013 26 / 27 Definition of a function √ 2 Find the domain of the function f (x) = x−1 x−3 Solution. Again, we must have x ≥ 1 but we must also prevent the denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞). Smith (SHSU) Elementary Functions 2013 26 / 27 Definition of a function √ 2 Find the domain of the function f (x) = x−1 x−3 Solution. Again, we must have x ≥ 1 but we must also prevent the denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞). Smith (SHSU) Elementary Functions 2013 26 / 27 Definition of a function √ 2 Find the domain of the function f (x) = x−1 x−3 Solution. Again, we must have x ≥ 1 but we must also prevent the denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞). Smith (SHSU) Elementary Functions 2013 26 / 27 Definition of a function √ 2 Find the domain of the function f (x) = x−1 x−3 Solution. Again, we must have x ≥ 1 but we must also prevent the denominator from being zero, so x cannot be 3, either. The domain is then all real numbers at least as big as 1 except for the number 3. Here is our answer in interval notation: The domain is [1, 3) ∪ (3, ∞). Smith (SHSU) Elementary Functions 2013 26 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27 Definition of a function √ 3 Find the domain of the function f (x) = x−1 x2 − 6x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x2 − 6x + 8 = (x − 2)(x − 4), so x cannot be 2 or 4. So our answer is all real numbers at least as big as 1 and not equal to 2 or 4. In interval notation, our answer is: The domain is [1, 2) ∪ (2, 4) ∪ (4, ∞.) (END) Smith (SHSU) Elementary Functions 2013 27 / 27