# 1.7 Inverse functions  (Page 3/10)

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When a function has no inverse function, it is possible to create a new function where that new function on a limited domain does have an inverse function. For example, the inverse of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)={x}^{2},\text{\hspace{0.17em}}$ because a square “undoes” a square root; but the square is only the inverse of the square root on the domain $\text{\hspace{0.17em}}\left[0,\infty \right),\text{\hspace{0.17em}}$ since that is the range of $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{x}.$

We can look at this problem from the other side, starting with the square (toolkit quadratic) function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}.\text{\hspace{0.17em}}$ If we want to construct an inverse to this function, we run into a problem, because for every given output of the quadratic function, there are two corresponding inputs (except when the input is 0). For example, the output 9 from the quadratic function corresponds to the inputs 3 and –3. But an output from a function is an input to its inverse; if this inverse input corresponds to more than one inverse output (input of the original function), then the “inverse” is not a function at all! To put it differently, the quadratic function is not a one-to-one function; it fails the horizontal line test, so it does not have an inverse function. In order for a function to have an inverse, it must be a one-to-one function.

In many cases, if a function is not one-to-one, we can still restrict the function to a part of its domain on which it is one-to-one. For example, we can make a restricted version of the square function $\text{\hspace{0.17em}}f\left(x\right)={x}^{2}\text{\hspace{0.17em}}$ with its range limited to $\text{\hspace{0.17em}}\left[0,\infty \right),\text{\hspace{0.17em}}$ which is a one-to-one function (it passes the horizontal line test) and which has an inverse (the square-root function).

If $\text{\hspace{0.17em}}f\left(x\right)={\left(x-1\right)}^{2}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[1,\infty \right),\text{\hspace{0.17em}}$ then the inverse function is $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)=\sqrt{x}+1.$

• The domain of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ = range of $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ = $\text{\hspace{0.17em}}\left[1,\infty \right).$
• The domain of $\text{\hspace{0.17em}}{f}^{-1}\text{\hspace{0.17em}}$ = range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ = $\text{\hspace{0.17em}}\left[0,\infty \right).$

Is it possible for a function to have more than one inverse?

No. If two supposedly different functions, say, $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h,\text{\hspace{0.17em}}$ both meet the definition of being inverses of another function $\text{\hspace{0.17em}}f,\text{\hspace{0.17em}}$ then you can prove that $\text{\hspace{0.17em}}g=h.\text{\hspace{0.17em}}$ We have just seen that some functions only have inverses if we restrict the domain of the original function. In these cases, there may be more than one way to restrict the domain, leading to different inverses. However, on any one domain, the original function still has only one unique inverse.

## Domain and range of inverse functions

The range of a function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is the domain of the inverse function $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$

The domain of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is the range of $\text{\hspace{0.17em}}{f}^{-1}\left(x\right).$

Given a function, find the domain and range of its inverse.

1. If the function is one-to-one, write the range of the original function as the domain of the inverse, and write the domain of the original function as the range of the inverse.
2. If the domain of the original function needs to be restricted to make it one-to-one, then this restricted domain becomes the range of the inverse function.

## Finding the inverses of toolkit functions

Identify which of the toolkit functions besides the quadratic function are not one-to-one, and find a restricted domain on which each function is one-to-one, if any. The toolkit functions are reviewed in [link] . We restrict the domain in such a fashion that the function assumes all y -values exactly once.

 Constant Identity Quadratic Cubic Reciprocal $f\left(x\right)=c$ $f\left(x\right)=x$ $f\left(x\right)={x}^{2}$ $f\left(x\right)={x}^{3}$ $f\left(x\right)=\frac{1}{x}$ Reciprocal squared Cube root Square root Absolute value $f\left(x\right)=\frac{1}{{x}^{2}}$ $f\left(x\right)=\sqrt[3]{x}$ $f\left(x\right)=\sqrt{x}$ $f\left(x\right)=|x|$

The constant function is not one-to-one, and there is no domain (except a single point) on which it could be one-to-one, so the constant function has no meaningful inverse.

The absolute value function can be restricted to the domain $\text{\hspace{0.17em}}\left[0,\infty \right),$ where it is equal to the identity function.

The reciprocal-squared function can be restricted to the domain $\text{\hspace{0.17em}}\left(0,\infty \right).$

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
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
William
what is f(x)=
I don't understand
Joe
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
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
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
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.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
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
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.
how do you find the period of a sine graph
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
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert