# 1.4 Inverse functions  (Page 6/11)

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$f\left(x\right)=\sqrt[3]{x-4}$

$f\left(x\right)={x}^{3}+1$

a. ${f}^{-1}\left(x\right)=\sqrt[3]{x-1}$ b. Domain: all real numbers, range: all real numbers

$f\left(x\right)={\left(x-1\right)}^{2},x\le 1$

$f\left(x\right)=\sqrt{x-1}$

a. ${f}^{-1}\left(x\right)={x}^{2}+1,$ b. Domain: $x\ge 0,$ range: $y\ge 1$

$f\left(x\right)=\frac{1}{x+2}$

For the following exercises, use the graph of $f$ to sketch the graph of its inverse function.

For the following exercises, use composition to determine which pairs of functions are inverses.

$f\left(x\right)=8x,g\left(x\right)=\frac{x}{8}$

These are inverses.

$f\left(x\right)=8x+3,g\left(x\right)=\frac{x-3}{8}$

$f\left(x\right)=5x-7,g\left(x\right)=\frac{x+5}{7}$

These are not inverses.

$f\left(x\right)=\frac{2}{3}x+2,g\left(x\right)=\frac{3}{2}x+3$

$f\left(x\right)=\frac{1}{x-1},x\ne 1,g\left(x\right)=\frac{1}{x}+1,x\ne 0$

These are inverses.

$f\left(x\right)={x}^{3}+1,g\left(x\right)={\left(x-1\right)}^{1\text{/}3}$

$f\left(x\right)={x}^{2}+2x+1,x\ge -1,\phantom{\rule{1em}{0ex}}g\left(x\right)=-1+\sqrt{x},x\ge 0$

These are inverses.

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

For the following exercises, evaluate the functions. Give the exact value.

${\text{tan}}^{-1}\left(\frac{\sqrt{3}}{3}\right)$

$\frac{\pi }{6}$

${\text{cos}}^{-1}\left(-\frac{\sqrt{2}}{2}\right)$

${\text{cot}}^{-1}\left(1\right)$

$\frac{\pi }{4}$

${\text{sin}}^{-1}\left(-1\right)$

${\text{cos}}^{-1}\left(\frac{\sqrt{3}}{2}\right)$

$\frac{\pi }{6}$

$\text{cos}\left({\text{tan}}^{-1}\left(\sqrt{3}\right)\right)$

$\text{sin}\left({\text{cos}}^{-1}\left(\frac{\sqrt{2}}{2}\right)\right)$

$\frac{\sqrt{2}}{2}$

${\text{sin}}^{-1}\left(\text{sin}\left(\frac{\pi }{3}\right)\right)$

${\text{tan}}^{-1}\left(\text{tan}\left(-\frac{\pi }{6}\right)\right)$

$-\frac{\pi }{6}$

The function $C=T\left(F\right)=\left(5\text{/}9\right)\left(F-32\right)$ converts degrees Fahrenheit to degrees Celsius.

1. Find the inverse function $F={T}^{-1}\left(C\right)$
2. What is the inverse function used for?

[T] The velocity V (in centimeters per second) of blood in an artery at a distance x cm from the center of the artery can be modeled by the function $V=f\left(x\right)=500\left(0.04-{x}^{2}\right)$ for $0\le x\le 0.2.$

1. Find $x={f}^{-1}\left(V\right).$
2. Interpret what the inverse function is used for.
3. Find the distance from the center of an artery with a velocity of 15 cm/sec, 10 cm/sec, and 5 cm/sec.

a. $x={f}^{-1}\left(V\right)=\sqrt{0.04-\frac{V}{500}}$ b. The inverse function determines the distance from the center of the artery at which blood is flowing with velocity V . c. 0.1 cm; 0.14 cm; 0.17 cm

A function that converts dress sizes in the United States to those in Europe is given by $D\left(x\right)=2x+24.$

1. Find the European dress sizes that correspond to sizes 6, 8, 10, and 12 in the United States.
2. Find the function that converts European dress sizes to U.S. dress sizes.
3. Use part b. to find the dress sizes in the United States that correspond to 46, 52, 62, and 70.

[T] The cost to remove a toxin from a lake is modeled by the function

$C\left(p\right)=75p\text{/}\left(85-p\right),$ where $C$ is the cost (in thousands of dollars) and $p$ is the amount of toxin in a small lake (measured in parts per billion [ppb]). This model is valid only when the amount of toxin is less than 85 ppb.

1. Find the cost to remove 25 ppb, 40 ppb, and 50 ppb of the toxin from the lake.
2. Find the inverse function. c. Use part b. to determine how much of the toxin is removed for $50,000. a.$31,250, $66,667,$107,143 b. $\left(p=\frac{85C}{C+75}\right)$ c. 34 ppb

[T] A race car is accelerating at a velocity given by

$v\left(t\right)=\frac{25}{4}t+54,$

where v is the velocity (in feet per second) at time t .

1. Find the velocity of the car at 10 sec.
2. Find the inverse function.
3. Use part b. to determine how long it takes for the car to reach a speed of 150 ft/sec.

[T] An airplane’s Mach number M is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $\mu =2{\text{sin}}^{-1}\left(\frac{1}{M}\right).$

Find the Mach angle (to the nearest degree) for the following Mach numbers.

1. $\mu =1.4$
2. $\mu =2.8$
3. $\mu =4.3$

a. $~92\text{°}$ b. $~42\text{°}$ c. $~27\text{°}$

[T] Using $\mu =2{\text{sin}}^{-1}\left(\frac{1}{M}\right),$ find the Mach number M for the following angles.

1. $\mu =\frac{\pi }{6}$
2. $\mu =\frac{2\pi }{7}$
3. $\mu =\frac{3\pi }{8}$

[T] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

$T\left(x\right)=5+18\phantom{\rule{0.1em}{0ex}}\text{sin}\left[\frac{\pi }{6}\left(x-4.6\right)\right],$

where $x$ is time in months and $x=1.00$ corresponds to January 1. Determine the month and day when the temperature is $21\text{°}\text{C}.$

$x\approx 6.69,8.51;$ so, the temperature occurs on June 21 and August 15

[T] The depth (in feet) of water at a dock changes with the rise and fall of tides. It is modeled by the function

$D\left(t\right)=5\phantom{\rule{0.1em}{0ex}}\text{sin}\left(\frac{\pi }{6}t-\frac{7\pi }{6}\right)+8,$

where $t$ is the number of hours after midnight. Determine the first time after midnight when the depth is 11.75 ft.

[T] An object moving in simple harmonic motion is modeled by the function

$s\left(t\right)=-6\phantom{\rule{0.1em}{0ex}}\text{cos}\left(\frac{\pi t}{2}\right),$

where $s$ is measured in inches and $t$ is measured in seconds. Determine the first time when the distance moved is 4.5 ft.

$~1.5\phantom{\rule{0.2em}{0ex}}\text{sec}$

[T] A local art gallery has a portrait 3 ft in height that is hung 2.5 ft above the eye level of an average person. The viewing angle $\theta$ can be modeled by the function

$\theta ={\text{tan}}^{-1}\frac{5.5}{x}-{\text{tan}}^{-1}\frac{2.5}{x},$

where $x$ is the distance (in feet) from the portrait. Find the viewing angle when a person is 4 ft from the portrait.

[T] Use a calculator to evaluate ${\text{tan}}^{-1}\left(\text{tan}\left(2.1\right)\right)$ and ${\text{cos}}^{-1}\left(\text{cos}\left(2.1\right)\right).$ Explain the results of each.

${\text{tan}}^{-1}\left(\text{tan}\left(2.1\right)\right)\approx -1.0416;$ the expression does not equal 2.1 since $2.1>1.57=\frac{\pi }{2}$ —in other words, it is not in the restricted domain of $\text{tan}\phantom{\rule{0.1em}{0ex}}x.\phantom{\rule{0.2em}{0ex}}{\text{cos}}^{-1}\left(\text{cos}\left(2.1\right)\right)=2.1,$ since 2.1 is in the restricted domain of $\text{cos}\phantom{\rule{0.1em}{0ex}}x.$

[T] Use a calculator to evaluate $\text{sin}\left({\text{sin}}^{-1}\left(-2\right)\right)$ and $\text{tan}\left({\text{tan}}^{-1}\left(-2\right)\right).$ Explain the results of each.

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