INVERSE OF ONE-TO-ONE FUNCTIONS || GRADE 11 GENERAL MATHEMATICS Q1
Summary
TLDRThis video script explains the concept of inverse functions in mathematics. It covers the criteria for a function to have an inverse, including being one-to-one, and demonstrates how to find the inverse of various functions through examples. It also discusses when a function does not have an inverse, such as quadratic functions, and concludes with a problem-solving example converting temperatures between Fahrenheit and Kelvin.
Takeaways
- π The inverse of a function reverses the input and output, swapping the domain and range of the original function.
- π A function has an inverse if and only if it is one-to-one, meaning each output corresponds to exactly one input.
- π To find the inverse of a function, you interchange x and y, then solve for y in terms of x.
- π The horizontal line test can be used to determine if a function is one-to-one; if any horizontal line intersects the graph more than once, it's not one-to-one.
- π’ For a function like y = 2x - 1, substituting values into the function and then reversing the x and y values helps find the inverse.
- β Examples are given to illustrate finding inverses, such as y = 3x + 1 leading to an inverse of y = (x - 1) / 3.
- π Quadratic functions are not one-to-one because their graphs are parabolas that do not pass the horizontal line test.
- π’ The absolute value function, like f(x) = |3x|, also fails the horizontal line test and thus does not have an inverse.
- π‘ Converting temperature scales involves understanding inverse functions, such as converting from Fahrenheit to Kelvin and vice versa.
- β Not all functions have inverses; if a function fails the horizontal line test, it does not have an inverse function.
- π To check if two functions are inverses, you can use function composition (f(g(x)) should equal x, and g(f(x)) should also equal x).
Q & A
What is the definition of an inverse function?
-An inverse function reverses the effect of the original function. If a function f performs a certain operation on x to produce y, then the inverse function undoes that operation to get back to x.
How is the domain and range of a function related to its inverse?
-The domain of the inverse function is the range of the original function, and the range of the inverse function is the domain of the original function.
What is the process of finding the inverse of a function?
-To find the inverse of a function, you first write the function in the form y = f(x), then interchange x and y, and finally solve for y in terms of x.
What is the condition for a function to have an inverse?
-A function has an inverse if and only if it is one-to-one, meaning each input is mapped to exactly one output.
How do you determine if a function is one-to-one?
-You can determine if a function is one-to-one by using the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is one-to-one.
What is the inverse of the function f(x) = 2x - 1?
-The inverse of the function f(x) = 2x - 1 is f^(-1)(x) = (x + 1) / 2.
How do you find the inverse of a function given by a set of ordered pairs?
-To find the inverse of a function given by a set of ordered pairs, you swap the x and y values of each pair.
What is the inverse of the function g(x) = x^3 - 2?
-The inverse of the function g(x) = x^3 - 2 is g^(-1)(x) = (x + 2)^(1/3).
Why can't a quadratic function have an inverse?
-A quadratic function cannot have an inverse because it fails the horizontal line test. A parabola intersects any horizontal line at most once, but an inverse function requires a unique input for each output.
How do you convert a temperature from Kelvin to Fahrenheit using the inverse function?
-To convert a temperature from Kelvin to Fahrenheit, you use the inverse function: F = (K - 273.15) * (9/5) + 32.
How can you check if two functions are inverses of each other?
-You can check if two functions are inverses of each other by composing them and seeing if the result is the identity function, which leaves the input unchanged.
Outlines
π Understanding Inverse Functions
This paragraph discusses the concept of inverse functions in mathematics. It explains that an inverse function reverses the effect of the original function, swapping the domain and range. The inverse is only possible for one-to-one functions, where each output is produced by exactly one input. The script uses the example of the function y = 2x - 1, demonstrating how to find its inverse by substituting y for x and solving for the new x, resulting in the inverse function x = (y + 1) / 2. The paragraph emphasizes that the inverse function's domain and range are the range and domain of the original function, respectively.
π Steps to Find the Inverse of a Function
The second paragraph outlines the steps to find the inverse of a given one-to-one function. It starts by expressing the function in the form y = f(x), then interchanges x and y, and finally solves for y in terms of x. The paragraph provides examples, such as finding the inverse of f(x) = 3x + 1, which results in the inverse function f^(-1)(x) = (x - 1) / 3. Another example is given for the function g(x) = x^3 - 2, where the inverse is found by solving for y and taking the cube root, resulting in g^(-1)(x) = cube root of (x + 2). The paragraph also explains that a function has an inverse if and only if it is one-to-one.
π’ Solving for Inverses of Rational Functions
This paragraph delves into finding the inverse of more complex functions, such as rational functions. It describes the process of solving for y in terms of x by cross-multiplication and rearranging terms. An example is given for the function f(x) = (2x + 1) / (3x - 4), where the inverse is found by interchanging x and y, and then solving for y to get f^(-1)(x) = (2x + 1) / (3x - 2). The paragraph also addresses why certain functions, like quadratic functions and those involving absolute values, do not have inverses because they fail the horizontal line test, meaning they are not one-to-one functions.
π‘ Converting Temperature Scales with Inverse Functions
The fourth paragraph applies the concept of inverse functions to convert temperature scales. It explains how to find the inverse function that converts from Kelvin to Fahrenheit, given the original function K(T) = (5/9)*T - 32 + 273.15. The process involves solving for T in terms of K, which results in the inverse function T(K) = 9/5*(K - 273.15) + 32. The paragraph also discusses how to determine if two functions are inverses of each other by using the composition of functions and checking if the result is the identity function.
π’ Engaging with the Audience
The final paragraph is a call to action for the audience to engage with the content by liking, subscribing, and hitting the bell button to stay updated with the channel's videos. It serves as a conclusion to the video script, encouraging viewers to interact with the channel for more educational content.
Mindmap
Keywords
π‘Inverse Function
π‘Domain and Range
π‘One-to-One Function
π‘Horizontal Line Test
π‘Substitution
π‘Interchange of Variables
π‘Absolute Value Function
π‘Composition of Functions
π‘Identity Function
π‘Quadratic Function
Highlights
Definition of inverse function and its relationship with the original function's domain and range.
Explanation of how to find the inverse of a function by swapping domain and range.
Example of finding the inverse function for a given set of ordered pairs.
Process of swapping x and y values to find the inverse function of a one-to-one function.
Detailed steps to find the inverse of a function using algebraic manipulation.
Example of finding the inverse of a linear function (3x + 1).
Example of finding the inverse of a cubic function (x^3 - 2).
Explanation of why a quadratic function cannot have an inverse function.
Example of finding the inverse of a rational function (2x + 1 / 3x - 4).
Explanation of how to handle absolute value functions and their inability to have inverses.
Conversion formula from Fahrenheit to Kelvin and its inverse from Kelvin to Fahrenheit.
How to verify if two functions are inverses of each other using composition.
Example of verifying inverse functions using composition for f(x) = -1/2x and g(x) = -2x.
Importance of one-to-one functions in having valid inverses.
Practical application of inverse functions in temperature conversion.
Transcripts
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that and an universe for example what is
the reverse of positivity negative good
but sub-process performed by function f
of X is called inverse of function X
this means that the domain of the
inverse is the rings of the original
function so rather than a pen non
dominant inverse is the range of the
original function as a as the definition
said reverse and that the range of the
inverse is the domain of the original
function so it bees are being class you
don't mean in rain-snow original
function but that means inverse McGrail
inverse lung in Delaware and don't mean
original function Ranger sir inverse you
arrange an original function domain
Chester inverse that and Danielle for
example we have the original function y
is equal to 2x minus 1 so for example my
entire table of values from negative 4
to positive 4 if we substitute negative
force to the original function the value
of y is negative 9 if we substitute
negative 3 to the original function the
values negative 7 ok same as negative 2
negative 1 0 1 2 3 4 so this is not the
body so why so kappa'd kakuni noting an
inverse so I know my IRA so much really
versatile not a new y that will be the
value of x and you X that will be the
value of y so in a pocket apollon saying
apology Megara how to reverse
the original function okay so let's say
this is your domain you know X naught in
the original function DT can dominate a
negative 4 negative 3 negative 2
negative 1 0 1 2 3 4
when I plug that in fact that things are
inverse u negative 4 negative 3 negative
2 negative 1 0
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inverse one-to-one function
let F be a one-to-one with domain a and
range B then the inverse of F denoted as
f place the negative 1 so this is the
symbol for inverse know it's a function
with domain be in range a so again
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defined by function or the inverse
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of X is equal to Y for any Y NP okay a
function has an inverse if and only if
it is one-to-one
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function inverting the x and y values of
a function result in a function if and
only if the original function is
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add numbering exponent okay so DARPA
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value Union possible the indicia
one-to-one
for example find the inverse of the
function described by the set of ordered
pairs we have 1 negative 3 2 & 1
3 3 4 5 5 7 so again I'm gonna win
inheritance which the party needs of
each ordered pair the original function
is that so a mulligan inverse function
at n will be negative 3 1 1 2 3 3 5 4 &
7 5
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one function bucket then you move along
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function is worth the word okay
to find the inverse of a one-to-one
function given the equation Maritime
Putnam sushant being steps first write
the function in the form y is equal to f
of X next interchange the X and y
variables next and is the last have
solved for Y in terms of X for example
number 1 find inverse of f of X is equal
to 3x plus 1 so that is the original
function first we need to change f of X
into y so muggin y equals 3x plus 1 so
finally turning that into f of x and y
so now it interchange the variable so
you why Kagami nothing X Y or X Gaga we
not in Y so magic in X is equal to 3 y
plus 1
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and not in muscle see why Lee but not in
C+ one D talk so mugging X minus 1
equals 3 y again I seen Assad not in
detail volume of Y so para Makua not in
your body now why we need to divide both
sides by 3 so therefore I'm dividing
both sides by 3 and so 3 y 2 by 3 I'm
not eternal my detour X minus 1 divided
over 3 so that is the value of y
therefore I use in a 10 so dabit morning
why are using ulong so that is now the
inverse of 3x plus 1 okay Lyla getting
it on a simple data pertaining thereto I
inverse so the inverse of X is equal to
X minus 1 over 3 that and any learning
steps may be nominated another example
find the inverse of G of X is equal to X
cubed minus 2 again your example a given
function X cubed minus 3 is a 1 to 1 so
therefore LM not enemy function my
inverse store again and gagawin k-pop
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interchange the variables you wipe up a
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and then solve for y now so our notion
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negative 2 it will become X plus 2 is
equal to Y cube a maritime cube no Marin
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cube Jan so I don't go in
by cubing metal toy t-cubed
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root of x plus two is equal to y cube
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three don't sigh mugging expose exponent
okay say you nobody nothing cube boom
shaka billah para my wallet on three so
therefore why now is a cube root of x
plus two penny will enter in parenthesis
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parenthesis cube root of x plus 2 again
but a mossad nut is y so he knew root
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retire so yeah
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you why do mean cube root of x plus 2
therefore the inverse of X cubed minus 2
is the cube root of x plus 2 okay
next find inverse of f of X is equal to
2x plus 1 over 3x minus 4 so rational
thought major
hababam solution the Gagarina 10 wooden
steps change nothing young FX naught and
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papa returning x human x naught in d top
operator not in my next step solve for y
in terms of x okay so cross
multiplication 2x times 3y minus 4 it
anion is equal to 2y plus 1 of course it
is distribute not into XD to sell of x
times 3y is 3x y minus x times 4 is 4x
is equal to 2y plus 1 since this is odd
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see a good group not insulin dalawa so
Stu Wiley Padma don't Panama Kazama
selling it on 3x why Satan a on 3x y -2
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plus 1 para Mis old nut is CY illa
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divided by 3x salamati Terra - copy the
sign - why / is equal to 2 equals copy
4x plus 1 again since why I am sin
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not in a 3x minus 2 he divided not and
both side by 3 X minus 2 so therefore
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2 + ma TT rainylin is why it on a man 4x
plus 1 divide 3x minus 2 so he turning
inverse not n so the inverse of function
2x plus 1 over 3x minus 4 is equal to 2x
plus 1 over 3x minus 4
ok so that will be the function again so
you in person at the end is 4x plus 1
over 3x minus 2 so Shannon game steps
the end ok so meconium inverse next
example
find the inverse of f of X is equal to x
squared plus 4x minus 2 first observe
the given function this function is
quadratic so Hopak quadratic the graph
is parabola pinging you bop opposition's
a one-to-one function using the
horizontal line test
hindi therefore this is a quadratic
function with a graph in the shape of
parabola that opens upward it is not a
one-to-one function as it feels no
result alike there so indeed nothing mr.
Sabzian persica seen is a one-to-one
function
okay so Tinga and observed nuan given
function Peres in dicta in XS ol next
find the inverse of f of X is equal to
the absolute value of 3x again absolute
valley and kappa Greenup not in your
absolute value that is letter V but I'll
be honest shape no graph and it fails
also in horizontal line test so the
graph of f of X is equal to absolute
value 3x is shaped like a B whose vertex
is located at the origin this function
fails the horizontal line test and
therefore has no inverse okay nothing in
my opinion I'll give you one problem
solving okay to convert from degrees
Fahrenheit to Kelvin the function is K
of T is equal to 5 over 9 times t minus
32 plus 273.15 where T is the
temperature in Fahrenheit kelvins the SI
unit of the temperature find inverse
function converting the temperature in
Kelvin to degrees Fahrenheit
garnet Z function okay we need to look
at the inverse okay
right the original function next okay so
I know many re after that
okay McGee kinky nailiang not in shadow
we represent K up tsk no Indian Gilligan
me tetanus why not lagging f of X when I
pretend I'm not in a white Saudi top
palette and anything and K so k minus
200 so it taught the 273.15 D but not in
dawn so mugging k minus 273.15 is equal
to 5 over 9 times t minus 32 next okay
since keno who are not in you inverse
okay you know who are nothing in inverse
so I know I'm money are in a on we need
to divide both side by what we need to
divide both side by seeking over why not
end right by not 5 over 9 okay divide
not in both sides by 5 over 9 CC Tunica
5 over 9 right okay so no money are a
after that you our equations but in
evite nothing and both side by 5 over 9
that will become what okay this one nah
okay so can you know what not in
universe vomit ah ha ha OH
next since quinoa nothing in tea not
only but nothing you - 32 disability
that will become 9 over 5 times K minus
273.15 plus 32 and the value of that is
9 of
five times K minus 273.15 plus 32 and
that is also now the inverse function oh
this original function so I thought
named in the end
no no name Indian sauna how to process
this problem Panama not in Malayalam on
okay
like this example determine whether the
function below are inverses of each
other or not for example my given time
function f of X is equal to negative 1/2
X and G of X is equal to negative 2x
pinocchio not in my element I turned 11
function at oh I inverse a sad-sack we
are going to use composition panel on so
you ex not in detour Papa returned not
innovation and G of X okay oh let say f
of X I'm public user axiom valanor G of
X so the main value no G of X negative 2
X so therefore young X new detour Papa
returned up in a negative 2x so my
concern not a new negative to negative 1
times negative X is positive x so habil
annamund u X nah man
none gee Papa little nothing of value
dang f of X which is negative 1/2 X so
therefore u X naught and Papa returned
nothing of value now f of X now negative
1 of X so Omar consoles in egged C
positive 2 so negative times negative
that is positive x kappa equal answer
good now using the composition
ibiza BN by daddy finish on the two
functions are inverses of each other
dot and I know
Gaga meeting in a composition Parimal a
man and a long functional inversely
lasat Issa and then kept an equal eundel
ah one function a on Ibiza bein in verse
Allah says erisa so salomina to tune and
KO about inverse of one-to-one function
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