# 1.4 Inverse functions  (Page 5/11)

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## The maximum value of a function

In many areas of science, engineering, and mathematics, it is useful to know the maximum value a function can obtain, even if we don’t know its exact value at a given instant. For instance, if we have a function describing the strength of a roof beam, we would want to know the maximum weight the beam can support without breaking. If we have a function that describes the speed of a train, we would want to know its maximum speed before it jumps off the rails. Safe design often depends on knowing maximum values.

This project describes a simple example of a function with a maximum value that depends on two equation coefficients. We will see that maximum values can depend on several factors other than the independent variable x .

1. Consider the graph in [link] of the function $y=\text{sin}\phantom{\rule{0.1em}{0ex}}x+\text{cos}\phantom{\rule{0.1em}{0ex}}x.$ Describe its overall shape. Is it periodic? How do you know?

Using a graphing calculator or other graphing device, estimate the $x$ - and $y$ -values of the maximum point for the graph (the first such point where x >0). It may be helpful to express the $x$ -value as a multiple of π.
2. Now consider other graphs of the form $y=A\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+B\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$ for various values of A and B . Sketch the graph when A = 2 and B = 1, and find the $x$ - and y -values for the maximum point. (Remember to express the x -value as a multiple of π, if possible.) Has it moved?
3. Repeat for A = 1, B = 2. Is there any relationship to what you found in part (2)?
4. Complete the following table, adding a few choices of your own for A and B :
A B x y A B x y
0 1 $\sqrt{3}$ 1
1 0 1 $\sqrt{3}$
1 1 12 5
1 2 5 12
2 1
2 2
3 4
4 3
5. Try to figure out the formula for the y -values.
6. The formula for the $x$ -values is a little harder. The most helpful points from the table are $\left(1,1\right),\left(1,\sqrt{3}\right),\left(\sqrt{3},1\right).$ ( Hint : Consider inverse trigonometric functions.)
7. If you found formulas for parts (5) and (6), show that they work together. That is, substitute the $x$ -value formula you found into $y=A\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+B\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$ and simplify it to arrive at the $y$ -value formula you found.

## Key concepts

• For a function to have an inverse, the function must be one-to-one. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test.
• If a function is not one-to-one, we can restrict the domain to a smaller domain where the function is one-to-one and then define the inverse of the function on the smaller domain.
• For a function $f$ and its inverse ${f}^{-1},f\left({f}^{-1}\left(x\right)\right)=x$ for all $x$ in the domain of ${f}^{-1}$ and ${f}^{-1}\left(f\left(x\right)\right)=x$ for all $x$ in the domain of $f.$
• Since the trigonometric functions are periodic, we need to restrict their domains to define the inverse trigonometric functions.
• The graph of a function $f$ and its inverse ${f}^{-1}$ are symmetric about the line $y=x.$

## Key equations

• Inverse functions
${f}^{-1}\left(f\left(x\right)\right)=x\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}x\phantom{\rule{0.2em}{0ex}}\text{in}\phantom{\rule{0.2em}{0ex}}D,\text{and}\phantom{\rule{0.2em}{0ex}}f\left({f}^{-1}\left(y\right)\right)=y\phantom{\rule{0.2em}{0ex}}\text{for all}\phantom{\rule{0.2em}{0ex}}y\phantom{\rule{0.2em}{0ex}}\text{in}\phantom{\rule{0.2em}{0ex}}R.$

For the following exercises, use the horizontal line test to determine whether each of the given graphs is one-to-one.

Not one-to-one

Not one-to-one

One-to-one

For the following exercises, a. find the inverse function, and b. find the domain and range of the inverse function.

$f\left(x\right)={x}^{2}-4,x\ge 0$

a. ${f}^{-1}\left(x\right)=\sqrt{x+4}$ b. Domain $\text{:}\phantom{\rule{0.2em}{0ex}}x\ge -4,\text{range}\text{:}\phantom{\rule{0.2em}{0ex}}y\ge 0$

maxima and minima problem in log form
find Maxima and minima of 4^x - 8xlog2
Roshan
(x-1)4^(x-1) - 8log2 4^(x - 1) x - 4^(x - 1) - 8 log(2) -1 to 3 or -11 to 13.
James
thanks
Roshan
The f'(4)for f(x) =4^x
I need help under implicit differentiation
how to understand this
nhie
understand what?
Boniface
how dont you understand it is obvious
Se
if it's there, it must be differentiable
Giorgio
marginal rate of substitution in economics for an example is an implicit differentiation function. It is a proportion of comparison. It could be expressed as the area of a triangle of the old value proportional to the new, and then the next value and so on.
James
The new shapes form a line with a derivative curve
James
That curve could be expressed mathematically. A good real life example is the proportion at which people barter in pawn shops. "How about 100?" "What are you trying to do? rob me? 50!" "No way chap. 75." "Ill give you 62. deal?" "68. Nothing more nothing less." "deal"
James
the proportion of what someone was trying to get for their product, versus what they were offered to the new price they wanted for their product and what they were offered
James
The proportion of the differentiating triangles would be somewhat 1:2. and since there is little variation to the curve then it looks more like a straight line.
James
By the way This is one of the hardest subjects for me. I have a really hard time expressing things in such a way. I'm trying to have more exact calculations which is why I still study the subject.
James
What is derivative of antilog x dx ?
what's the meaning of removable discontinuity
what's continuous
Brian
an area under a curve is continuous because you are looking at an area that covers a range of numbers, it is over an interval, such as 0 to 4
Lauren
using product rule x^3,x^5
Kabiru
may god be with you
Sunny
Luke 17:21 nor will they say, See here or See there For indeed, the kingdom of God is within you. You've never 'touched' anything. The e-energy field created by your body has pushed other electricfields. even our religions tell us we're the gods. We live in energies connecting us all. Doa/higgsfield
Scott
if you have any calculus questions many of us would be happy to help and you can always learn or even invent your own theories and proofs. math is the laws of logic and reality. its rules are permanent and absolute. you can absolutely learn calculus and through it better understand our existence.
Scott
ya doubtless
Bilal
help the integral of x^2/lnxdx
Levis
also find the value of "X" from the equation that follow (x-1/x)^4 +4(x^2-1/x^2) -6=0 please guy help
Levis
Use integration by parts. Let u=lnx and dv=x2dx Then du=1xdx and v=13x3. ∫x2lnxdx=13x3lnx−∫(13x3⋅1x)dx ∫x2lnxdx=13x3lnx−∫13x2dx ∫x2lnxdx=13x3lnx−19x3+C
Bilal
itz 1/3 and 1/9
Bilal
now you can find the value of X from the above equation easily
Bilal
Pls i need more explanation on this calculus
usman
usman from where do you need help?
Levis
thanks Bilal
Levis
integrate e^cosx
Uchenna
-sinx e^x
Leo
Do we ask only math question? or ANY of the question?
yh
Gbesemete
How do i differentiate between substitution method, partial fraction and algebraic function in integration?
usman
you just have to recognize the problem. there can be multiple ways to solve 1 problem. that's the hardest part about integration
Lauren
test
we asking the question cause only the question will tell us the right answer
Sunny
find integral of sin8xcos12xdx
don't share these childish questions
Bilal
well find the integral of x^x
Levis
bilal kumhar you are so biased if you are an expert what are you doing here lol😎😎😂😂 we are here to learn and beside there are many questions on this chat which you didn't attempt we are helping each other stop being naive and arrogance so give me the integral of x^x
Levis
Levis I am sorry
Bilal
Bilal it okay buddy honestly i am pleasured to meet you
Levis
x^x ... no anti derivative for this function... but we can find definte integral numerically.
Bilal
thank you Bilal Kumhar then how we may find definite integral let say x^x,3,5?
Levis
evaluate 5-×square divided by x+2 find x as limit approaches infinity
i have not understood
Leo
Michael
welcome
Sunny
I just dont get it at all...not understanding
Michagaye
0 baby
Sunny
The denominator is the aggressive one
Sunny
wouldn't be any prime number for x instead ?
Harold
or should I say any prime number greater then 11 ?
Harold
just wondering
Harold
I think as limit Approach infinity then X=0
Levis
ha hakdog hahhahahaha
ha hamburger
Leonito
Fond the value of the six trigonometric function of an angle theta, which terminal side passes through the points(2x½-y)²,4
What's f(x) ^x^x
What's F(x) =x^x^x
Emeka
are you asking for the derivative
Leo
that's means more power for all points
rd
Levis
iam sorry f(x)=x^x it means the output(range ) depends to input(domain) value of x by the power of x that is to say if x=2 then x^x would be 2^2=4 f(x) is the product of X to the power of X its derivatives is found by using product rule y=x^x introduce ln each side we have lny=lnx^x =lny=xlnx
Levis
the derivatives of f(x)=x^x IS (1+lnx)*x^x
Levis
So in that case what will be the answer?
Alice
nice explanation Levis, appreciated..
Thato
what is a maximax
A maxima in a curve refers to the maximum point said curve. The maxima is a point where the gradient of the curve is equal to 0 (dy/dx = 0) and its second derivative value is a negative (d²y/dx² = -ve).
Viewer
what is the limit of x^2+x when x approaches 0
it is 0 because 0 squared Is 0
Leo
0+0=0
Leo
simply put the value of 0 in places of x.....
Tonu
the limit is 2x + 1
Nicholas
the limit is 0
Muzamil
limit s x
Bilal
The limit is 3
Levis
Leo we don't just do like that buddy!!! use first principle y+∆y=x+∆x ∆y=x+∆x-y ∆y=(x+∆x)^2+(x+∆x)-x^2+x on solving it become ∆y=3∆x+∆x^2 as ∆x_>0 limit=3 if you do by calculator say plugging any value of x=0.000005 which approach 0 you get 3
Levis
find derivatives 3√x²+√3x²
3 + 3=6
mujahid