Download Definition of a function and more Exams Mathematics in PDF only on Docsity!
Elementary Functions
Part 1, Functions Lecture 1.1a, The Definition of a Function
Dr. Ken W. Smith
Sam Houston State University
Smith (SHSU) Elementary Functions 2013 1 / 27
Definition of a function
We study the most fundamental concept in mathematics, that of a function. In this lecture we first define a function and then examine the domain of functions defined as equations involving real numbers.
Definition of a function. A function f : X → Y assigns to each element of the set X an element of Y. Picture a function as a machine,
Smith (SHSU) Elementary Functions 2013 2 / 27
A function machine
We study the most fundamental concept in mathematics, that of a function. In this lecture we first define a function and then examine the domain of functions defined as equations involving real numbers.
Definition of a function A function f : X → Y assigns to each element of the set X an element of Y. Picture a function as a machine,
droppingSmith x (SHSU)-values into one end of the machine and picking up Elementary Functions y-values at 2013 3 / 27
Inputs and unique outputs of a function
The set X of inputs is called the domain of the function f. The set Y of all conceivable outputs is the codomain of the function f. The set of all outputs is the range of f. (The range is a subset of Y .)
The most important criteria for a function is this:
A function must assign to each input a unique output.
We cannot allow several different outputs to correspond to an input.
Smith (SHSU) Elementary Functions 2013 4 / 27
Examples of functions
We give an example (from Wikipedia) of a function from a set X to the set Y.
The function maps 1 to D, 2 to C and 3 to C.
Note that each element of X has a unique output in Y.
Smith (SHSU) Elementary Functions 2013 5 / 27
Not a function
However the map below is not a function.
Some items in X are not mapped anywhere; worse, the item 2 has two outputs, both B and C.
Functions are not allowed to change a single input into several outputs!
Smith (SHSU) Elementary Functions 2013 6 / 27
Functions as questions
Functions occur naturally in our world.
When we pull out an attribute of an object, we are essentially creating a function.
For example, the set X below has polygons with various colors. The question, “What is the color of a polygon?” could be viewed as a function that maps to polygons to colors.
Smith (SHSU) Elementary Functions 2013 7 / 27
SSN and Sam ID as functions
Functions occur throughout our modern technological society.
The US social security number is a function SSN mapping US citizens to nine digit numbers.
At Sam Houston State University, all students and staff are assigned a Sam ID.
This as a function SamID, mapping students/staff to nine digit numbers.
For example,
SamID(Ken W Smith) = 000354765.
(This function exists so that data about students/staff – classes, grades, wages, etc. – can be kept in a computer database, tracked by a single number.)
Smith (SHSU) Elementary Functions 2013 8 / 27
Elementary Functions
Part 1, Functions Lecture 1.1b, Functions defined by equations
Dr. Ken W. Smith
Sam Houston State University
Smith (SHSU) Elementary Functions 2013 13 / 27
Functions defined by equations
Many functions we explore in mathematics and science are defined by an equation. We can define a function implicitly in an equation involving two variables. For example, does the equation
2 x + 3y − 4 = 0
define a function with inputs x and outputs y? Isolate y to get
3 y = 4 − 2 x and so y =
(4 − 2 x).
We may now explicitly define the function
f (x) =
(4 − 2 x).
So YES, the equation 2 x + 3y − 4 = 0 does define a function. Smith (SHSU) Elementary Functions 2013 14 / 27
Independent and dependent variables
A digression. When we considered the equation 2 x + 3y − 4 = 0 our choice of x as input and y as output is arbitrary. We could decide (contrary to custom!) that y is the input and x is the output. Then, solving for x, we have 2 x = 4 − 3 y and so x =
(4 − 3 y) and so we create the function g(y) =
(4 − 3 y).
But most of the time we will stick to convention and, unless stated otherwise, assume x is the input variable and y is the output variable. The input variable x is often called the independent variable and the output variable is the dependent variable since its value depends on the input. (^) Smith (SHSU) Elementary Functions 2013 15 / 27
Exercises on implicit functions
Some worked exercises.
1 Does the equation x^2 y = 4 define y as a function of x? (If it does, give the domain of the implied function.) Solution. We attempt to solve for y. We may multiply both sides of the equation by
x^2
as long as x is not zero. This gives us y =
x^2
Is there a problem with x = 0? No, x = 0 does not allow x^2 y = 4, so x will never be zero in this equation.
Answer: YES, this is a function; y =
x^2
The domain of this function is all real numbers except zero. In interval notation the domain is (−∞, 0) ∪ (0, ∞).
Smith (SHSU) Elementary Functions 2013 16 / 27
Not a function
2 Does the equation xy^2 = 4 define y as a function of x? Solution. If we attempt to solve for y, we multiply both sides of the equation by
x
(as long as x 6 = 0) and so we have y^2 =
x
But now, what is y? y could be positive or negative – there will generally be two choices here, one positive and one negative.
The appearance of two answers violates the uniqueness requirement in our outputs for a function.
Answer: NO, this is not a function. If x = 1 then we don’t know if y = 2 or y = − 2.
Smith (SHSU) Elementary Functions 2013 17 / 27
Not a function
3 Does the equation x^2 y = 0 define y as a function of x? (Why/why not?) Solution. Although it might be tempting to solve for y, first notice that if x is zero then y could be 0 or 1 or 2. 71828 or anything!
So the input x = 0 does not give a unique output. This is not a function.
Answer: NO; if x = 0 then y could be anything.
(This is different than problem 1. In problem 1, x = 0 is not a possible input in the equation. But here x = 0 is a possibility for a solution to the equation! So we have to worry about the input x.)
Smith (SHSU) Elementary Functions 2013 18 / 27
Practicing function notation
Let us practice the function notation, f (x). A formula for f (x) tells us how the input x leads to the output f (x). For example, suppose f (x) = x^2 − 9. Compute: 1 f (0), 2 f (1), 3 f (−1), 4 f (−5), 5 f (−x) Solutions. If f (x) = x^2 − 9 then 1 f (0) = 0^2 − 9 = − 9. 2 f (1) = (1)^2 − 9 = 1 − 9 = − 8. 3 f (−1) = (−1)^2 − 9 = 1 − 9 = − 8 , 4 f (−5) = (−5)^2 − 9 = 25 − 9 = 16. 5 f (−x) = (−x)^2 − 9 = x^2 − 9. Smith (SHSU) Elementary Functions 2013 19 / 27
Practicing function notation
More examples. Let’s continue with the function f (x) = x^2 − 9. Compute: 6 f (x + h), 7 f (
x), 8 f (2a + 1), 9 −f (x) + 2 Solutions. If f (x) = x^2 − 9 then 6 f (x + h) = (x + h)^2 − 9 = (x^2 + 2xh + h^2 ) − 9 = x^2 + 2xh + h^2 − 9 , 7 f (
x) =(
x)^2 − 9 = x − 9 ,
8 f (2a + 1) = (2a + 1)^2 − 9 = (4a^2 + 4a + 1) − 9 = 4 a^2 + 4a − 8 , 9 −f (x) + 2 = −(x^2 − 9) + 2 = −x^2 + 11.
Smith (SHSU) Elementary Functions 2013 20 / 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
3 Find the domain of the function f (x) =
x − 1 x^2 − 6 x + 8 Solution. We must have x ≥ 1 and we must prevent the denominator from being zero. The denominator factors as x^2 − 6 x + 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
