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f ( x ) = x 3 + 1

a. f −1 ( x ) = x 1 3 b. Domain: all real numbers, range: all real numbers

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f ( x ) = ( x 1 ) 2 , x 1

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f ( x ) = x 1

a. f −1 ( x ) = x 2 + 1 , b. Domain: x 0 , range: y 1

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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 ( x ) = 8 x , g ( x ) = x 8

These are inverses.

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f ( x ) = 8 x + 3 , g ( x ) = x 3 8

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f ( x ) = 5 x 7 , g ( x ) = x + 5 7

These are not inverses.

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f ( x ) = 2 3 x + 2 , g ( x ) = 3 2 x + 3

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f ( x ) = 1 x 1 , x 1 , g ( x ) = 1 x + 1 , x 0

These are inverses.

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f ( x ) = x 3 + 1 , g ( x ) = ( x 1 ) 1 / 3

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f ( x ) = x 2 + 2 x + 1 , x −1 , g ( x ) = −1 + x , x 0

These are inverses.

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f ( x ) = 4 x 2 , 0 x 2 , g ( x ) = 4 x 2 , 0 x 2

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

sin ( cos −1 ( 2 2 ) )

2 2

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sin −1 ( sin ( π 3 ) )

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tan −1 ( tan ( π 6 ) )

π 6

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The function C = T ( F ) = ( 5 / 9 ) ( F 32 ) converts degrees Fahrenheit to degrees Celsius.

  1. Find the inverse function F = T −1 ( C )
  2. What is the inverse function used for?
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[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 ( x ) = 500 ( 0.04 x 2 ) for 0 x 0.2 .

  1. Find x = f −1 ( V ) .
  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 ( V ) = 0.04 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

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A function that converts dress sizes in the United States to those in Europe is given by D ( x ) = 2 x + 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.
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[T] The cost to remove a toxin from a lake is modeled by the function

C ( p ) = 75 p / ( 85 p ) , 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. ( p = 85 C C + 75 ) c. 34 ppb

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

v ( t ) = 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.
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[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 μ = 2 sin −1 ( 1 M ) .

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

An image of a birds eye view of an airplane. Directly in front of the airplane is a sideways “V” shape, with the airplane flying directly into the opening of the “V” shape. The “V” shape is labeled “mach wave”. There are two arrows with labels. The first arrow points from the nose of the airplane to the corner of the “V” shape. This arrow has the label “velocity = v”. The second arrow points diagonally from the nose of the airplane to the edge of the upper portion of the “V” shape. This arrow has the label “speed of sound = a”. Between these two arrows is an angle labeled “Mach angle”. There is also text in the image that reads “mach = M > 1.0”.
  1. μ = 1.4
  2. μ = 2.8
  3. μ = 4.3

a. ~ 92 ° b. ~ 42 ° c. ~ 27 °

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[T] Using μ = 2 sin −1 ( 1 M ) , find the Mach number M for the following angles.

  1. μ = π 6
  2. μ = 2 π 7
  3. μ = 3 π 8
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[T] The temperature (in degrees Celsius) of a city in the northern United States can be modeled by the function

T ( x ) = 5 + 18 sin [ π 6 ( x 4.6 ) ] ,

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

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

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[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 ( t ) = 5 sin ( π 6 t 7 π 6 ) + 8 ,

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

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

s ( t ) = −6 cos ( π t 2 ) ,

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 sec

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[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 θ can be modeled by the function

θ = tan −1 5.5 x tan −1 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.

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[T] Use a calculator to evaluate tan −1 ( tan ( 2.1 ) ) and cos −1 ( cos ( 2.1 ) ) . Explain the results of each.

tan −1 ( tan ( 2.1 ) ) 1.0416 ; the expression does not equal 2.1 since 2.1 > 1.57 = π 2 —in other words, it is not in the restricted domain of tan x . cos −1 ( cos ( 2.1 ) ) = 2.1 , since 2.1 is in the restricted domain of cos x .

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[T] Use a calculator to evaluate sin ( sin −1 ( −2 ) ) and tan ( tan −1 ( −2 ) ) . Explain the results of each.

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Questions & Answers

f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
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Chicken nuggets
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MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
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find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
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Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
A Function F(X)=Sinx+cosx is odd or even?
WIZARD Reply
neither
David
Neither
Lovuyiso
f(x)=1/1+x^2 |=[-3,1]
Yuliana Reply
apa itu?
fauzi
determine the area of the region enclosed by x²+y=1,2x-y+4=0
Gerald Reply
Hi
MP
Hi too
Vic
hello please anyone with calculus PDF should share
Adegoke
Which kind of pdf do you want bro?
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Usman
okay I have such documents
Fitzgerald
please share it
Hamza
How to use it to slove fraction
Tricia Reply
Hello please can someone tell me the meaning of this group all about, yes I know is calculus group but yet nothing is showing up
Shodipo
You have downloaded the aplication Calculus Volume 1, tackling about lessons for (mostly) college freshmen, Calculus 1: Differential, and this group I think aims to let concerns and questions from students who want to clarify something about the subject. Well, this is what I guess so.
Jean
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how can we scatch a parabola graph
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Endalkachew
how can I solve differentiation?
Sir Reply
with the help of different formulas and Rules. we use formulas according to given condition or according to questions
CALCULUS
For example any questions...
CALCULUS
v=(x,y) وu=(x,y ) ∂u/∂x* ∂x/∂u +∂v/∂x*∂x/∂v=1
log tan (x/4+x/2)
Rohan
what is the procedures in solving number 1?
Vier Reply
Practice Key Terms 5

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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