INVERSE FUNCTION // GENERAL MATHEMATICS // TAGALOG
Summary
TLDRThe video explains how to find the inverse of various mathematical functions step by step. It begins by substituting 'y' for 'f(x)' and then interchanging 'x' and 'y' in the equation. The video demonstrates solving for 'y' through examples involving linear, rational, and cubic functions, and touches on more complex cases like quadratic functions. For each example, the instructor walks through the process of solving the equation, cross-multiplying, simplifying, and factoring to isolate 'y', ultimately determining the inverse function.
Takeaways
- 🔄 The first step in finding an inverse function is to change f(x) to y.
- ↔️ After that, swap x and y in the equation to find the inverse function.
- ➗ Solve for y by isolating it on one side of the equation and simplifying.
- ➕ Example 1: The inverse of f(x) = 3x + 6 is (x - 6) / 3.
- ➕ Example 2: The inverse of f(x) = 2x + 3 is (x - 3) / 2.
- ➕ Example 3: For a more complex function like f(x) = (3x - 7) / (4x + 3), you use cross-multiplication to solve.
- 💡 Use cross-multiplication when the function is a fraction to eliminate denominators.
- 🧮 Factor out common terms when simplifying expressions to find the inverse.
- 🔎 Always check for steps like removing fractions by multiplying both sides by the denominator.
- 🧩 Cube root functions require applying the cube root to both sides to isolate y.
Q & A
What is the first step in finding the inverse function?
-The first step is to change f(x) into y, so the equation becomes y = f(x).
After changing f(x) to y, what is the next step?
-The next step is to switch the roles of x and y, so x = f(y), and then solve for y.
How do you isolate y in the equation x = 3y + 6?
-To isolate y, first subtract 6 from both sides to get x - 6 = 3y. Then, divide both sides by 3 to get y = (x - 6)/3.
What does the inverse function represent in this context?
-The inverse function represents the function that 'undoes' the original function, mapping outputs back to inputs.
How do you find the inverse of a function when fractions are involved, such as f(x) = (3x - 7)/(4x + 3)?
-Start by switching x and y, then cross-multiply to eliminate the fraction, and solve for y using algebraic methods.
Why is cross-multiplication necessary when solving for the inverse of a rational function?
-Cross-multiplication helps eliminate the denominator, simplifying the equation so that you can isolate y and solve for the inverse.
How do you deal with cube roots when finding the inverse of functions like f(x) = x^3 + 5?
-To find the inverse, switch x and y to get x = y^3 + 5, then subtract 5 from both sides, and take the cube root of both sides to isolate y.
When dealing with quadratic functions, such as f(x) = x^2 - 2x + 3, what special technique is often used?
-Completing the square is a common technique used to simplify quadratic equations when finding their inverses.
What does it mean to 'complete the square' when finding the inverse of a quadratic function?
-Completing the square involves rewriting the quadratic equation in the form of (x + a)^2 to simplify solving for y.
Why might some inverse functions include both positive and negative solutions?
-Some inverse functions, especially those involving squares or cube roots, can have both positive and negative solutions due to the nature of the operations involved.
Outlines
🔍 Understanding How to Find the Inverse of a Function
In this section, the process of finding the inverse function is explained. The first step is converting f(x) to y and then swapping x and y. After solving for y, the inverse function formula is derived. The example used involves f(x) = 3x + 6, where the inverse function is determined to be (x - 6) / 3. Another example uses f(x) = 2x + 3, leading to an inverse of (x - 3) / 2.
➗ Working with Rational Functions to Find Inverses
This section focuses on a more complex example, f(x) = (3x - 7) / (4x + 3), and shows how to find its inverse. The steps involve cross-multiplication, distribution, and combining like terms. After some algebraic manipulation, the inverse function is determined as (3x + 7) / (3 - 4x). Another rational function example, f(x) = (4x + 7) / (2x - 3), follows a similar process, resulting in an inverse function of (-3x - 7) / (2(2 - x)).
🔄 Finding the Inverse of Cubic and Fractional Functions
This part covers the inverse function for cubic and fractional functions like f(x) = x^3 + 5 and f(x) = (1/2)(3x + 4). The method involves switching x and y, solving for y, and taking cube roots for cubic equations. For fractional functions, fractions are cleared by multiplying both sides by the denominator. The inverse functions are derived as cube root(x - 5) and (2x - 4) / 3, respectively.
⚖️ Solving Inverse Functions for Cubic Roots and Quadratics
This paragraph demonstrates finding inverse functions for more challenging equations like f(x) = cube root(x - 5) and f(x) = cube root(1 - 4x). The process includes switching x and y, solving by taking cubes, and manipulating the algebra. The results include inverse functions such as (x^3 - 1) / -4. Finally, the quadratic equation f(x) = x^2 - 2x + 3 is introduced, which involves completing the square to find its inverse.
🧮 Applying Completing the Square to Find Inverse of Quadratics
The final section covers a quadratic function, f(x) = x^2 + 2x - 3. The focus is on the technique of completing the square to rewrite the quadratic function in a form that allows solving for y. After performing the algebraic steps, including finding square roots and rearranging terms, the inverse function is found as ±sqrt(x + 2) - 1, highlighting the complexity of finding inverses for quadratic functions.
Mindmap
Keywords
💡Inverse Function
💡Solving for y
💡Cross Multiply
💡Cube Root
💡Completing the Square
💡Quadratic Function
💡Algebraic Manipulation
💡Swapping x and y
💡Distributive Property
💡Factoring
Highlights
First step in finding the inverse function: change f(x) into y.
After changing f(x) into y, swap x and y in the equation.
Solve for y by isolating it on one side of the equation.
Example 1: For f(x) = 3x + 6, the inverse function is (x - 6) / 3.
For f(x) = 2x + 3, the inverse function is (x - 3) / 2.
For a more complex function like f(x) = (3x - 7) / (4x + 3), cross multiplication is used to find the inverse.
Distribute and combine like terms when solving for y in rational functions.
Factor the left-hand side when isolating y to solve for the inverse.
The inverse of f(x) = (3x - 7) / (4x + 3) is (3x + 7) / (3 - 4x).
For quadratic functions, completing the square may be required to find the inverse.
Handling cube root functions: Solve by isolating the cube root and then taking the cube of both sides.
Example 2: For f(x) = x³ + 5, the inverse function is the cube root of (x - 5).
Example 3: For f(x) = cube root of (1 - 4x), inverse is (x³ - 1) / -4.
Handling square root functions requires squaring both sides of the equation to remove the root.
Inverse functions for quadratics may involve factoring and solving for y with positive and negative square roots.
Transcripts
okay this time guys let us discuss on
how to find the inverse function the
first step that we are going to do
is we have to change this f of x into y
so
i'm giving y then bring down attention
3x plus six
then after that young x governating y
ating y governating x so i'm gonna
give x which is equal to three y plus
six
then after that solve for the value of y
so lipatnates
positive six
x minus six which is equal to three y
then after that angle going out and
solve for the value of y by dividing
both sides by three
by three so therefore this is one so
this is y which is equal to x minus six
all over three so mere nothing inverse
function which is are represented by
this the inverse function
is equal to
x minus six
all over three so i don't know i think
inverse function
supported not an expression i think
final answer guys
step or the the first one on how to find
the inverse function
let us proceed the second one
f of x naught n is equal to two x plus
three
first step and f of x governance uh
shang y
then we have here
bring down two x plus three epoch
palette notice x and y so i'm gigging
x which is equal to 2y plus 3.
then solve for the value of x so region
but not as positive 3 so x minus 3 is
equal to 2y then solve for the value of
y
so divide by dividing both sides by 2 so
2 divided by 2 cancel so this is y which
is equal to
x minus 3
over two
so marinating volume why not in lindito
so therefore the inverse nothing
of a function is
so this is x minus three
over two so this is now the inverse
function at n
okay let us have here another example we
have here f of x is equal to three x
minus seven all over four x plus three
so what we are going to do so bring down
log not n or
which is equal to bring down attention
3x minus 7 all over 4x plus 3.
then you probably don't ncx and y so
i
x which is equal to
three y minus seven all over four y
plus three
then what we're going to do is cross
multiply my imaginary values i determine
a one
so one times 3y minus 7
is just equal to
3y minus 7.
x dominating
multiply number 9
x multiplied by
4y plus three
then distribute netenyon x naught is a
four y plus three so bring down language
three y minus seven
which is equal to shazam for x y so
distribute not n
is three times x is three x
so next logarithmic examination left
side
volume y
so bring down attending
three y so this is three y
[Music]
negative 4 x y
then bring down attention three x so
this is three x
libertas
negative seven magnitude positive seven
then after that factor nothing is a left
side so on gcfnia is y quantity of
um three minus four x
which is equal to three x plus seven
then
after that to get the value of y divide
both sides by
three
minus four x three minus four x and
three minus four x
so cancel
which is equal to
3x plus 7
all over 3 minus 4x
inverse function so which is in the form
of 3x plus 7 all over 3 minus 4x
so this is our inverse function
next manaten
is so we have here f of x is equal to
four x plus seven all over two x minus
three so same as usual f of x
governating y
which is equal to bring down four x plus
seven
all over two x minus three
then epoch particular n c y and x so i'm
going x which is equal to
four y plus seven
all over
two y minus three
then cross multiply not n so my
imaginary value then a one so one times
four y
so this is four y
plus seven
then x naught n multiplied by d so x
naught n multiplying attention
two y minus three
then after that distribute net n bring
down one another four y
plus seven which is equal to x times two
y so two x y
then x times negative three negative
three x
then after that combine like terms latin
value number y
so this is four y bring down attenuation
for y
then it on two x y
is
negative 2xy
which is equal to this is negative 3x
7
is making negative 7
then factor not 10
gcf nothing data is 2y so this is 2y
quantity of
[Music]
2
minus
x which is equal to negative three x
minus seven
so divide nothing both sides by
two two
multiplied by two minus x two minus x
and 2
quantity of 2 minus x
so in this case
with the same thing
is y which is equal to negative 3x
minus 7
all over
2
quantity of
2 minus
x
so this is x
then of course this is now the inverse
function at n young f function nothing
or the inverse function
is equal to
negative 3 x minus 7
all over 2 quantity of 2
minus x so this is now our
inverse function
okay another one so we have here f of x
is equal to one half quantity of four x
plus four
and f of x is equal to x cubed plus five
find the inverse so same as usual this
will become equal to y which is equal to
one half then three x plus four
then after that guys um change the x and
y so it is making x and one half
quantity of three y plus four
then after that uh remove nothing young
fraction at n so but angle at the top
fraction
one half by multiplying by two because
two times one half is equal to one so
multiplying along negative both sides by
two
by two so it is two times x is two x
then
uh two times one half is one
one times three y plus four is three y
plus four so net angle
uh fractional one half so therefore
solve for the value of y natalya so this
is two x then libat
positive 4 negative 4 which is equal to
3y
then solve for the value of y so this is
now uh 3 3.
so
here y value not n is
2x or 2x minus 4 all over 3. so
therefore the inverse function at n
is equal to
x minus 4
all over three
so this is now our
answer
so how about this one a man so f of x is
equal to x cubed plus five so same
routine so this will become y which is
equal to x cubed plus five but palette
now that is x and y so i'm giving x
which is equal to y cubed plus five
then
uh yeah lipid latency positive five mug
again negative five so x minus five is
equal to y cube
so my problem that i did though it is
not a y cube so indeed number nine
putting it divide the n by
cube
okay so angle coming at n is we have to
uh multiply both sides by its cube root
so cube root not n
multiplied nothing by the cube root
so therefore the answer is the cube root
of
x minus 5.
so therefore our inverse function
the inverse function now is not the one
x raised to negative 1
of x is equal to cube root of
x minus five
okay
so this will be your final answer
another example so we have here f of x
is equal to the cube root of x minus
five so same as usual so this is y is
equal to cube root of x minus five
okay
then after that i'll attend x and y so x
is equal to the cube root of y
minus five
then of course um tangalyn cube root by
multiplying by
its cube okay multiply nothing both
sides uh cube multiply nothing both
sides uh cube so it is
x cube which is equal to canceling
attento so this is y
minus five
so
bring down attenuation
x cubed plus five
okay so this is now your inverse
function the inverse function
is now equal to
x cubed plus five
almost the same here sakabila so we have
here f of x is equal to the cube root of
one minus four x so same as usual one
y rather is equal to the cube root of
one minus four x
then interchange so this is x is equal
to the cube root
of one minus four y
then after that multiply both sides by
cube
by cube
so this cube so cancel cancel d is x
cube which is
um
equal to 1 minus 4 y
so this time so we have here
uh positive one magnitude negative one
which is equal to negative four y
since
uh we have here negative four y divide
both sides by negative four negative 4
so this is now equal to
y which is equal to x cubed minus 1 all
over negative 4.
so here our inverse function
is now equal to
x cubed minus 1 all over negative 4.
so this is now our
inverse function
okay so we have here another example f
of x is equal to x squared minus two x
plus three so autonomous in a form of
quadratic then so measure my contain
difference
of quadratic nato so same as usual
so bring down an attention y which is
equal to
x squared plus
two x
minus three
okay so in this case uh
usually
x is equal to
y square
plus
uh
2y
then minus three thousand origin this is
x minus three
tables up four plus three
the opposite is y square
plus
two y
tables
is y quantity of y
plus
two
so it is x plus three suppose divide
both both sides by y plus two y plus two
so on the origin is young y naught
on this kind of quadratic so this time
you apply nothing you're completing the
square or maybe process the entire
parameter
is bring down muna nathan completing the
square taiyo what monetary interchange
number x and y
y
which is equal to
bring down national x squared
then plus 2x
was data but completing the square tire
so completing the square i'm gonna go
nothing
which is one cinerle
one
square nothing
then
negative one square
minus three
so theta simplify monetize y is equal to
x plus one
square
tables
um
negative one square is positive one
then minus three
simplify so this y is equal to
quantity of x plus one
square
one minus three is negative two
so lipo
um negative two
or d times interchange
y so y
plus one
square
minus two
so atom ion
is lipid is negative two
so x
plus two
is equal to
y plus one
so in this case
my squared i d though
so
nothing so by multiplying both sides by
its roots so
so this is a positive negative so
positive negative square root of
x plus two so which is equal to kinase
so this is y
plus one so nipple
your
positive one similarly
negative one
square root of positive negative
na
x plus 2
so this is your y value so inverse
function at n
and inverse function at n
is ethanol which is the negative 1
plus negative square root of
x plus two
okay so this is our inverse function
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