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3.7 Inverse functions  (Page 5/9)

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Using the graph in [link] , (a) find   g − 1 ( 1 ) , and (b) estimate   g − 1 ( 4 ) .

a. 3; b. 5.6

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Finding inverses of functions represented by formulas

Sometimes we will need to know an inverse function for all elements of its domain, not just a few. If the original function is given as a formula—for example,   y   as a function of   x — we can often find the inverse function by solving to obtain   x   as a function of   y .

Given a function represented by a formula, find the inverse.

  1. Make sure   f   is a one-to-one function.
  2. Solve for   x .
  3. Interchange   x   and   y .

Inverting the fahrenheit-to-celsius function

Find a formula for the inverse function that gives Fahrenheit temperature as a function of Celsius temperature.

C = 5 9 ( F − 32 )
C = 5 9 ( F − 32 ) C ⋅ 9 5 = F − 32 F = 9 5 C + 32

By solving in general, we have uncovered the inverse function. If

C = h ( F ) = 5 9 ( F − 32 ) ,

then

F = h − 1 ( C ) = 9 5 C + 32

In this case, we introduced a function   h   to represent the conversion because the input and output variables are descriptive, and writing   C − 1   could get confusing.

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Solve for   x   in terms of   y   given   y = 1 3 ( x − 5 ) .

x = 3 y + 5

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Solving to find an inverse function

Find the inverse of the function   f ( x ) = 2 x − 3 + 4.

y = 2 x − 3 + 4 Set up an equation . y − 4 = 2 x − 3 Subtract 4 from both sides . x − 3 = 2 y − 4 Multiply both sides by  x − 3  and divide by  y − 4. x = 2 y − 4 + 3 Add 3 to both sides .

So   f − 1 ( y ) = 2 y − 4 + 3   or   f − 1 ( x ) = 2 x − 4 + 3.

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Solving to find an inverse with radicals

Find the inverse of the function   f ( x ) = 2 + x − 4 .

y = 2 + x − 4 ( y − 2 ) 2 = x − 4 x = ( y − 2 ) 2 + 4

So   f − 1 ( x ) = ( x − 2 ) 2 + 4.

The domain of   f   is   [ 4 , ∞ ) .   Notice that the range of   f   is   [ 2 , ∞ ) ,   so this means that the domain of the inverse function   f − 1   is also   [ 2 , ∞ ) .

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What is the inverse of the function   f ( x ) = 2 − x ? State the domains of both the function and the inverse function.

f − 1 ( x ) = ( 2 − x ) 2 ;     domain     of     f :     [ 0 , ∞ ) ;     domain     of     f − 1 :     ( − ∞ , 2 ]

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Finding inverse functions and their graphs

Now that we can find the inverse of a function, we will explore the graphs of functions and their inverses. Let us return to the quadratic function   f ( x ) = x 2   restricted to the domain   [ 0 , ∞ ) , on which this function is one-to-one, and graph it as in [link] .

Graph of f(x).
Quadratic function with domain restricted to [0, ∞).

Restricting the domain to   [ 0 , ∞ )   makes the function one-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.

We already know that the inverse of the toolkit quadratic function is the square root function, that is, f − 1 ( x ) = x . What happens if we graph both f   and f − 1 on the same set of axes, using the x - axis for the input to both f  and   f − 1 ?

We notice a distinct relationship: The graph of   f − 1 ( x )   is the graph of   f ( x )   reflected about the diagonal line   y = x ,   which we will call the identity line, shown in [link] .

Graph of f(x) and f^(-1)(x).
Square and square-root functions on the non-negative domain

This relationship will be observed for all one-to-one functions, because it is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

Finding the inverse of a function using reflection about the identity line

Given the graph of   f ( x )   in [link] , sketch a graph of   f − 1 ( x ) .

Graph of f^(-1)(x).

This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of   ( 0 , ∞ )   and range of   ( − ∞ , ∞ ) ,   so the inverse will have a domain of   ( − ∞ , ∞ )   and range of   ( 0 , ∞ ) .

If we reflect this graph over the line   y = x ,   the point   ( 1 , 0 )   reflects to   ( 0 , 1 )   and the point   ( 4 , 2 )   reflects to   ( 2 , 4 ) .   Sketching the inverse on the same axes as the original graph gives [link] .

Graph of f(x) and f^(-1)(x).
The function and its inverse, showing reflection about the identity line
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Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
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what is a capacitor?
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Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
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please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
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velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
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50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
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I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
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about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
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Felix Reply
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ALIYU
field is a region of space under the influence of some physical properties
Collete
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Another formula for Acceleration
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a=v/t. a=f/m a
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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