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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 , ) .

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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.

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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.
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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).
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Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
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Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
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Modress
-x=7
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deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
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solve for me this equational y=2-x
Rubben Reply
what are you solving for
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solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
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Rubben
it will b -y+2=x
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how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
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Stephen Reply
What is Indices
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If one side only of a triangle is given is it possible to solve for the unkown two sides?
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Practice Key Terms 1

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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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