Download Elementary Functions - Sam Houston State University

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