Is there any function that is equal to its own inverse?
Yes. If
$\text{\hspace{0.17em}}f={f}^{-1},\text{\hspace{0.17em}}$ then
$\text{\hspace{0.17em}}f\left(f\left(x\right)\right)=x,\text{\hspace{0.17em}}$ and we can think of several functions that have this property. The identity function does, and so does the reciprocal function, because
$$\frac{1}{\frac{1}{x}}=x$$
Any function
$\text{\hspace{0.17em}}f\left(x\right)=c-x,\text{\hspace{0.17em}}$ where
$\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is a constant, is also equal to its own inverse.
Access these online resources for additional instruction and practice with inverse functions.
Visit
this website for additional practice questions from Learningpod.
Key concepts
If
$\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ is the inverse of
$\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ then
$\text{\hspace{0.17em}}g(f(x))=f(g(x))=x.\text{\hspace{0.17em}}$ See
[link] ,
[link] , and
[link] .
Each of the toolkit functions has an inverse. See
[link] .
For a function to have an inverse, it must be one-to-one (pass the horizontal line test).
A function that is not one-to-one over its entire domain may be one-to-one on part of its domain.
For a tabular function, exchange the input and output rows to obtain the inverse. See
[link] .
The inverse of a function can be determined at specific points on its graph. See
[link] .
To find the inverse of a formula, solve the equation
$\text{\hspace{0.17em}}y=f(x)\text{\hspace{0.17em}}$ for
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ as a function of
$\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Then exchange the labels
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ See
[link] ,
[link] , and
[link] .
The graph of an inverse function is the reflection of the graph of the original function across the line
$\text{\hspace{0.17em}}y=x.\text{\hspace{0.17em}}$ See
[link] .
Section exercises
Verbal
Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?
Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that
$\text{\hspace{0.17em}}y$ -values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no
$\text{\hspace{0.17em}}y$ -values repeat and the function is one-to-one.
How do you find the inverse of a function algebraically?
Given a function
$\text{\hspace{0.17em}}y=f(x),\text{\hspace{0.17em}}$ solve for
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ in terms of
$\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Interchange the
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ Solve the new equation for
$\text{\hspace{0.17em}}y.\text{\hspace{0.17em}}$ The expression for
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ is the inverse,
$\text{\hspace{0.17em}}y={f}^{-1}(x).$
Show that the function
$\text{\hspace{0.17em}}f(x)=a-x\text{\hspace{0.17em}}$ is its own inverse for all real numbers
$\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$
For the following exercises, find a domain on which each function
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ restricted to that domain.
$f(x)={(x+7)}^{2}$
domain of
$f(x):\text{\hspace{0.17em}}[-7,\infty );\text{\hspace{0.17em}}{f}^{-1}(x)=\sqrt{x}-7$
Given
$\text{\hspace{0.17em}}f\left(x\right)=\frac{x}{2}+x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g(x)=\frac{2x}{1-x}:$
Find
$\text{\hspace{0.17em}}f(g(x))\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g(f(x)).$
What does the answer tell us about the relationship between
$\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g(x)?$
a.
$\text{\hspace{0.17em}}f(g(x))=x\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g(f(x))=x.\text{\hspace{0.17em}}$ b. This tells us that
$\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ are inverse functions
I've run into this:
x = r*cos(angle1 + angle2)
Which expands to:
x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2))
The r value confuses me here, because distributing it makes:
(r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1))
How does this make sense? Why does the r distribute once
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis
vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As
'f(x)=y'.
According to Google,
"The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks.
"Â" or 'Â' ... Â
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Period =2π
if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts