<< Chapter < Page Chapter >> Page >

The domain of function f is ( 1 , ) and the range of function f is ( −∞ , −2 ) . Find the domain and range of the inverse function.

The domain of function f 1 is ( , 2 ) and the range of function f 1 is ( 1 , ) .

Got questions? Get instant answers now!

Finding and evaluating inverse functions

Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases.

Inverting tabular functions

Suppose we want to find the inverse of a function represented in table form. Remember that the domain of a function is the range of the inverse and the range of the function is the domain of the inverse. So we need to interchange the domain and range.

Each row (or column) of inputs becomes the row (or column) of outputs for the inverse function. Similarly, each row (or column) of outputs becomes the row (or column) of inputs for the inverse function.

Interpreting the inverse of a tabular function

A function f ( t ) is given in [link] , showing distance in miles that a car has traveled in t minutes. Find and interpret f 1 ( 70 ) .

t  (minutes) 30 50 70 90
f ( t )  (miles) 20 40 60 70

The inverse function takes an output of f and returns an input for f . So in the expression f 1 ( 70 ) , 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function f , 90 minutes, so f 1 ( 70 ) = 90. The interpretation of this is that, to drive 70 miles, it took 90 minutes.

Alternatively, recall that the definition of the inverse was that if f ( a ) = b , then f 1 ( b ) = a . By this definition, if we are given f 1 ( 70 ) = a , then we are looking for a value a so that f ( a ) = 70. In this case, we are looking for a t so that f ( t ) = 70 , which is when t = 90.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using [link] , find and interpret (a)   f ( 60 ) , and (b)   f 1 ( 60 ) .

t  (minutes) 30 50 60 70 90
f ( t )  (miles) 20 40 50 60 70
  1. f ( 60 ) = 50. In 60 minutes, 50 miles are traveled.
  2. f 1 ( 60 ) = 70. To travel 60 miles, it will take 70 minutes.
Got questions? Get instant answers now!

Evaluating the inverse of a function, given a graph of the original function

We saw in Functions and Function Notation that the domain of a function can be read by observing the horizontal extent of its graph. We find the domain of the inverse function by observing the vertical extent of the graph of the original function, because this corresponds to the horizontal extent of the inverse function. Similarly, we find the range of the inverse function by observing the horizontal extent of the graph of the original function, as this is the vertical extent of the inverse function. If we want to evaluate an inverse function, we find its input within its domain, which is all or part of the vertical axis of the original function’s graph.

Given the graph of a function, evaluate its inverse at specific points.

  1. Find the desired input on the y -axis of the given graph.
  2. Read the inverse function’s output from the x -axis of the given graph.

Evaluating a function and its inverse from a graph at specific points

A function g ( x ) is given in [link] . Find g ( 3 ) and g 1 ( 3 ) .

Graph of g(x).

To evaluate g ( 3 ) , we find 3 on the x -axis and find the corresponding output value on the y -axis. The point ( 3 , 1 ) tells us that g ( 3 ) = 1.

To evaluate g 1 ( 3 ) , recall that by definition g 1 ( 3 ) means the value of x for which g ( x ) = 3. By looking for the output value 3 on the vertical axis, we find the point ( 5 , 3 ) on the graph, which means g ( 5 ) = 3 , so by definition, g 1 ( 3 ) = 5. See [link] .

Graph of g(x).
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
modul questions trigonometry
Thamarai Reply
(1+cosA)(1-cosA)=sin^2A
BINCY
differentiate f(t)=1/4t to the power 4 +8
Jessica Reply
I need trigonometry,polynomial
duru Reply
ok
Augustine
Why is 7 on top
Bertha Reply
simplify cot x / csc x
Catherine Reply
👉🌹Solve🌻 Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
what is the period of cos?
SIYAMTHEMBA Reply
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
Patrick
simplify: cot x/csc x
Catherine
sorry i didnt realize you were actually asking someone else to put their question on here. i thought this was where i was supposed to.
Catherine
some to dereve formula for bulky density
kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
questions
Thamarai
ok
AjA
Practice Key Terms 1

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask