Functions
Summary
TLDRThis educational video script introduces the concept of functions in mathematics, explaining how to identify if a mapping is a function using mapping diagrams and the vertical line test. It covers one-to-one and many-to-one functions, function notation, and how to find the range and domain of a function. The script also discusses excluding values from the domain, composite functions, and finding inverse functions. It provides examples and step-by-step instructions for understanding and applying these mathematical concepts.
Takeaways
- 🔍 To determine if a mapping is a function, ensure each element in the domain maps to exactly one element in the range.
- 📊 Mapping diagrams and the vertical line test are useful tools to visualize whether a relationship is a function.
- 📐 A one-to-one function is where each element in the domain maps to a unique element in the range, while a many-to-one function has multiple elements in the domain mapping to the same element in the range.
- ❌ Diagrams where elements in the domain map to more than one element in the range, or vice versa, do not represent functions.
- 📝 Function notation, like \( f(x) \), is used to represent the rule by which an input is transformed into an output.
- 🔢 The domain of a function includes all possible input values, while the range consists of all possible output values.
- ⛔ Certain values, like zero in division or negative numbers in square roots, may need to be excluded from the domain as they lead to undefined operations.
- 🔄 Composite functions are created when one function's output becomes the input for another function, with the order affecting the final result.
- 🔄 The notation \( g(f(x)) \) indicates that function \( f \) is applied first, followed by function \( g \).
- 🔙 Inverse functions reverse the action of the original function; if \( f \) adds one, \( f^{-1} \) subtracts one.
- 📉 To find the inverse of a function, rewrite the function as \( y = \), swap \( x \) and \( y \), solve for \( y \), and use the notation \( f^{-1}(x) \).
Q & A
What is a function in mathematical terms?
-A function is a mathematical relationship where each element of one set, called the domain, is associated with exactly one element of another set, called the range.
How can you determine if a mapping is a function using a mapping diagram?
-A mapping is a function if each member of the first set (domain) is connected by exactly one arrow to an element of the second set (range), with no element in the second set being connected to more than one arrow.
What is the difference between a one-to-one function and a many-to-one function?
-A one-to-one function is where each element in the domain maps to a unique element in the range, and each element in the range is mapped to by exactly one element from the domain. A many-to-one function allows multiple elements in the domain to map to the same element in the range.
How do you use the vertical line test to determine if a graph represents a function?
-The vertical line test checks if any vertical line drawn on the graph intersects more than once with the graph. If it intersects only at one point for every vertical line, then the graph represents a function.
What is function notation and how is it used?
-Function notation is a way to represent a function using a symbol, often a letter like 'f'. For example, if 'f(x)' represents a function, then 'f(5) = 10' means that when the input '5' is put into the function 'f', the output is '10'.
Can you explain the concept of domain and range in the context of functions?
-The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). The domain is what you put into the function, and the range is what you get out of it.
Why are some values excluded from the domain of a function?
-Values are excluded from the domain of a function if they lead to impossible operations, such as division by zero or taking the square root of a negative number, which are undefined in real number arithmetic.
What is a composite function and how is it formed?
-A composite function is formed when one function is followed by another. For example, if you have functions f(x) and g(x), then the composite function g(f(x)) means you first apply f to x, then apply g to the result of f(x).
How do you find the composite function (g ∘ f)(x) given functions f(x) and g(x)?
-To find the composite function (g ∘ f)(x), you first evaluate f(x) to get an output, then use that output as the input for g(x). The result is the value of (g ∘ f)(x).
What is an inverse function and how can you determine if a function has an inverse?
-An inverse function is a function that reverses the effect of the original function. If applying the original function and then its inverse to a number yields the original number, and vice versa, then the function has an inverse. The notation for the inverse of a function f is f^(-1)(x).
How do you find the inverse of a given function algebraically?
-To find the inverse of a function algebraically, you first write the function as y = f(x), then swap x and y to get x = f(y), and finally solve for y to express it in terms of x. This new expression is the inverse function, denoted as f^(-1)(x).
Outlines
📐 Introduction to Functions
The script begins with an introduction to the concept of functions in mathematics. It outlines the objectives of the lesson, which include understanding how to determine if a mapping is a function, using function notation, finding the range of a function, identifying values that may need to be excluded from the domain, finding composite functions, and determining inverse functions. The explanation emphasizes the importance of one-to-one and many-to-one relationships in functions and uses mapping diagrams to illustrate these concepts. It explains how a function can be identified by ensuring that each element of the first set maps to exactly one element of the second set. The script also introduces the vertical line test as a method to determine if a graph represents a function.
🔢 Function Notation and Operations
This section delves into function notation, explaining how a function is a set of rules for transforming one number into another. It uses the example of a doubling function, denoted as 'F', to demonstrate how function notation works. The script clarifies that if 'x' is the input, then '2x' is the output for the doubling function. It further explains how functions operate on all inputs and provides an example of a function that doubles and adds one. The concept of domain and range is introduced, with domain being the set of possible inputs and range being the set of possible outputs. The script also touches on values that may be excluded from the domain due to impossibilities such as division by zero or the square root of a negative number.
🔄 Composite Functions
The script moves on to discuss composite functions, which are created when one function is followed by another. It uses a graphical example to illustrate how the output changes based on the order in which functions are applied. The explanation includes a step-by-step process for finding composite functions, emphasizing the importance of the order of operations. The script also explains how the domain of one function (G) must align with the range of another function (F) when creating composite functions. It provides a practical example of finding the result of a composite function, 'f(g(x))', and clarifies the notation used for composite functions.
🔄 Inverse Functions
The final section of the script introduces inverse functions, which are functions that undo the operations of another function. It explains that if a function adds one, its inverse would subtract one. The script provides an example of two functions, F and G, and demonstrates how their inverses work. It outlines a step-by-step process for finding the inverse of a function, which includes writing the function as 'y =', swapping x and y, and solving for y to obtain the inverse function. The script concludes with an example of finding the inverse of a given function, 'f(x) = 6x + 4', and demonstrates how to apply the steps to arrive at the inverse function, 'f^(-1)(x) = (x - 4) / 6'.
Mindmap
Keywords
💡Function
💡Mapping Diagram
💡One-to-One Function
💡Many-to-One Function
💡Domain
💡Range
💡Composite Function
💡Inverse Function
💡Vertical Line Test
💡Function Notation
Highlights
Introduction to the concept of functions and their properties.
Explanation of how to determine if a mapping is a function using a mapping diagram.
Definition of one-to-one and many-to-one functions with examples.
Use of the vertical line test to identify whether a graph represents a function.
Introduction to function notation and how to use it to represent a set of rules.
Example of how to find the range of a function using a graph.
Explanation of domain and range in the context of functions.
Discussion on values that may need to be excluded from the domain of a function.
Introduction to composite functions and how they are formed.
Example of how to calculate a composite function step by step.
Explanation of how to find the inverse of a function.
Step-by-step guide on how to derive the inverse function from a given function.
Graphical representation of an inverse function and its relationship to the original function.
Practical example of finding the inverse of a specific function.
Summary of the process for finding the inverse function, including changing variables and solving for y.
Final remarks on the importance of understanding functions, their properties, and operations.
Transcripts
hello everyone now we are going to our
new topic Which is
functions so at the end of this lesson
you will be able to determine whether a
mapping is a
function use function notation find the
range of a
function understand Which values may
need to be exclud from a domain find a
composite functions and last find
inverse
functions okay so first we need to
identify
functions relationship that are one to
one and many to one are functions a
mapping diagram makes it easy to desde
if it is a function or not If only one
arrow Leaves each member of the set then
the relationship is a function for
example look at these two
diagrams first diagram is diagram
a a is a function as each Element of the
first set maps to exactly one Element of
the second set it is a one to one
function As You can see one maps to two
two maps to three and three maps to
F diagram B is a function as each
Element of the first set maps to exactly
one Element of the second set since the
Elements in the second set have more
than Element From the first set map to
them It is called a many to one function
for example you can see that in diagram
B min2 and2 is maps to
F so they are called many to
One OK now we have diagram c and diagram
d c is not a function as each Element of
the first set maps to more than one
Element of the second set as you can see
in diagram c one maps to min1 and 1 that
means diagram c is not a
function now diagram D D is not function
either as at least one Element of the
first set maps to more than one Element
of the second set as you can see in
diagram d one maps to two and three So
it means diagram D is not a
function we can also use the vertical
Line test to decide if the graph Shows a
function or not it will only intersect
at one point if it is a function state
if it is one to one or many to one for
example look at this graph Wherever The
Red vertical line is placed on the graph
It will only intersect at one point
showing for example that one
1.5 maps to
2.5 So this is a
function it is an example of a one to
one
function next is function
notation a function is a set of rules
for turning one number into
Another You can see on the graph if we
put a number into a function It will
produce another
number so a letter can be used to
represent the rule if we call the
doubling function F Then if we Put in 5
to the function It will produce
10 F has operated on 5 to give 10 so we
write F of 5 equ to
10 if X is input then 2x is output that
is the doubling
function so we write f x equs to 2x or
we can use this notation
okay so in
graphic you can see that if we Put in x
2 f it will produce 2x
Ok
next a function operates on all of the
input if the function is double and add
One Then if 4x is input 8X + 1 is output
So if we Put in
4x to a function and the function is
double so we need to double 4x 4x get
double we get
8X and add one the function says so the
result the output of this function is
s + 1 topic is domain and
range okay so look at this
graphic the only the only Numbers The
function can use are one two F and seven
so we called The ers as
domain and set produced by the function
is 3 f si n so we call these numbers as
range so basically domain is the input
The Numbers that we Put in a into a
function and range is the the result we
get or the output we get from a
function so a graph of a function gives
a useful picture of the domain and range
for example these two graph you can see
that X Axis is the domain and yaxis is
the range so the domain cor
to the xxis and the range corresponds to
the
yxis we continue to values excluded from
the
domain some numbers cannot be used for
the domain as they lead to Impossible
operations these operations are usually
division by Zer we cannot Divide a
number by Zer or the square root of a
negative number there are no square root
of a negative
number ok so for example the function is
1/ X as I
can as I already ST before we cannot
Divide a number
exud
x from the domain of
F second example
is
FX root of
X the square root of a negative number
is not allow though is possible to soter
So
X smaller than Zero must be excluded
from the domain of
F next topic is composite
function so when one function is
followed by another the result is a
composite function
look at the graph if fx = 2x and GX = x
+ 3 then when we Put in two into F We've
got F because 2* 2 is 4 And if this
result this 4 we Put into G We've got
seven because
because 4 + 3 e=
7 if the Order of this functions is
change then the output is different for
example when we P in two to G first
We've got 5 because 2 + 3 is 5 and after
it we Put in 5 into F We've got 10
because 2 times 5 is
10 So if X is input then when we Put in
X to F We've got F of X after we we Put
in F of X to G weve got G of F of
x g of
FX is usually written without the square
brackets as
GF of X GF of X means do F first then
followed by G note that the domain of g
is the range of F in the same way f g of
X means do G first and then for follow
by
F for example weve got fx = x s and GX =
x +
2 and the question ask us to find f
G3 so as i state before
fgx means do G first so we need to do g
g of th
first that means we Put in 3 to the g
function so weve got 3+ 2 Which is 5
after We did
G3 we need to put in the result the
output of the G3 to the F function So it
become F of
5 so f squ i mean so 5 squ is equal to
25 so the result of f g of3 is
25 we continue to inverse
function the inverse function Andes
whatever has been done by the
function so So if the function is add
one then the inverse function is
subtract
one these functions are FX for example
FX Div by x + 1 add 1 and GX is x- 1
subtract 1 So if F is followed by G then
whatever number is in
is also the output because the G
function
und Everything that the the F
function
done So if X is the input then X is also
the
output so the function g is called The
the inverse of the function f the
inverse of F is the function that
und whatever F has done the notation f-
1 is used for the inverse of
F note that graphically f-1
X is a Reflection of f x in y equ
x but what if the inverse function is
not obvious
Oke So this following steps will help
you to find the inverse
function step one write the function as
y
equ Step Two change any X to y and any y
to x and last step is to make y subject
giving the inverse function and then use
the correct f- 1 x notation for
example find invers of the function
given
fx= 6x +
4 okay so the first step we need to do
is y the function as y equ what what we
got
fx 6x + 4 so we need to the function as
y e= 6x +
4 Next step two is to change any X to y
and change any y to
X so before we have y e= 6x + 4 and we
need to change y to x and X to y so
We've got X Equals to 6y +
4 last step is to make y the sub giving
the inverse function and then we use the
correct
f-1 X notation so before we
already do x equ= to 6 Y + 4 so to make
y The Subject we need to move this F To
The Other Side so weve got x- 4 equ to
6y and to remove the 6 from 6y we need
to
Divide so weve got Y = x- 4 Div 6 or y =
x-
4/6 after that We just need to change
the y to the f- 1x so the F Min So the
f- 1x is equal to x- 4// 6 so the
inverse function of fx = 6x + 4 is
f- 1x = x- 4 Div
6 oh
[Musik]
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