<< 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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
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