how to find the domain of an inverse function

Domain, Range and Inverse

In the previous section we introduced a lot of new information.  In this section we are going to accept a closer await at a few of import concepts that might become lost.  That is, every bit we movement through the topic of functions, we will focus on some new properties, notation and ideas.  Inverse functions, domain and range oftentimes get forgotten by students.  That is, students forget the basics which makes taking the next steps very difficult.  So, this department is a chance for you to develop a conceptual understanding of these important, and often elusive, ideas.  This way you will understand and if you lot understand, yous're more than likely to remember in the future.

Let'due south work through some examples that will reinforce your understanding of iv things.

1. What a function is
   1. The difference between a relation and a function
2. Finding the domain of a function
3. Finding the range of a role
4. Finding the inverse of a function

Other than the get-go objective, these get quite complicated and nuanced as yous move through higher levels of mathematics.  Similar to how the Order of Operations isn't just a topic, but is in all of mathematics, functions are used throughout mathematics.  When you learn Calculus, functions are a huge part!  Domain and range, and inverse functions are massively important concepts.  If we tin can learn them now, when things are simpler, it's less of a jump later on.

Example one:

1. Is the relation shown a function?  Explain your reasoning.
2. Detect the domain and range.
3. If the relation is a part, find its inverse role (if possible).

Number ane

Not all relations are functions, but all functions are relations.  A relation but compares ii things in math.  A few examples are y = ten, or y < x.

A relation is a part if each input has exactly one output.  Do you know why y < x is Non a part?   How many values of y would be true if x was five?  Infinitely many.  That'southward more than i output (y) for one input (ten).

There are special functions chosen One to One functions.  A Ane to One office has a unique output for each input.  In other words, in that case, an output cannot be repeated.

The graph has an pointer on peak, which ways that the pattern continues.  Do you run across the design betwixt the coordinates shown?  It seems like this might be y = 10 ², just there are no values of ten less than zero!

Each input shown has only 1 output, and the arrow suggests that the blueprint continues.  Then, this is a function.

Number 2

Domain is the fix of all possible inputs.  This can be tricky to determine.  Sometimes the easier question is, "Is there a value for ten that does not have an output for this office?"

The graph doesn't show anything less than zippo for x.  While the graphed curve is increasing, it is also slightly pointing to the correct, which means it will continue having larger values for ten.  So the domain here is all numbers greater than or equal to cipher.

Note on Note:  There are different styles and notation structures used to list domain and range.  We volition not exist discussing those even so.  The focus here is on developing the agreement of the concept of domain and range, and how to find each in uncomplicated cases.

Range is the gear up of all possible outputs.  To determine this from a graph nosotros demand to look at the set of all of the y – coordinates on the graph.  The smallest y – coordinate, called the minimum, is 0.  The graph is increasing from there, so the range is all numbers greater than or equal to zilch.

Domain: x ≥ 0

Range: y ≥ 0.

Number 3

A function's human relationship between the input and output is inverted for the function's changed role.  There is a lot going on with inverse functions, as there is with domain and range.  In time to come sections we will get deep and more technical with functions, only our goal hither is to develop a working agreement.  It is too easy to get lost in the procedures and technicalities later and lose sight of the pregnant.

If g(A) = B, that is the input value of A, into the function named g, produced an output of B, and another function produced the output of A, from the input B, these could be inverse functions.  It would have to be the cases that the input and outputs were inverted for the ii functions for all possible outputs, and there's a way to confirm this that you lot'll learn in a time to come section on inverse functions.  But, the primal concept is that the input/output human relationship is reversed for the inverse role.

Let's wait at our set of inputs and outputs, and make a listing of the integer coordinates.

For our inverse office, we'd have to swap the input and output.

Is there a office that has this relationship?  What about

Our original function is actually  with the domain intentionally restricted to be x ≥ 0.  Our inverse part is

with the outputs restricted to be greater than or equal to zero.

Example two:

1. Is the relation a function? Explicate your reasoning.
2. Find the domain and range.
3. If the relation is a function, observe its changed function (if possible).

Number 1

This relation is a function because each input has exactly 1 output.

Number 2

Here the domain is express.  In the previous example, we had a curve. That means there were infinitely many inputs and outputs.  Here, we have just four inputs and three outputs.

The domain is -two, 0, iii, 11.  (Putting them in club by size is a good practise.)

The range is 4, xi, 17.

Number 3

To find the inverse nosotros switch the input and output.  Nosotros'd end up with the following.

Do y'all see that the changed is NOT a function?  The input (to the inverse role) of iv has 2 outputs.  The reason this happened is because the role was non a One to One function.

Example 3:

1. Is the relation a part?  Explain your reasoning.
2. Observe the domain.

Number i

This is a office considering x – five has just one reply for any single input value nosotros cull for x.  Three divided by whatever number has only 1 reply.  Then each input has exactly one output.

Let'southward talk almost dividing by 3.  Is there whatsoever number that can cause bug when dividing?

Number 2

Division with the number null is tricky.  Allow's make certain we're solid here.  Segmentation is a question about multiplication.  For case, the reason 15/3 = five is considering 5 × 3 = fifteen.

Then, 8 ÷ 4 is really asking a question:  What times four is 8.

That's why this is naught:

And this is undefined:

There is nothing times zero that is equal to 5.

If at that place is a value for x, that makes the denominator equal to naught, then nosotros have a value missing from our domain.  Practise you see that if x = five, this part is undefined?  That means, the domain is every Real Number except 5.

We skipped the range on this instance considering it involves math beyond what you're able to approach, at this fourth dimension.  It isn't too complicated, but involves a lot of new ideas that are foreign to y'all correct now.  When the time comes, this will be easily understood.

We besides skipped the changed of this function, for now.  In this course y'all volition learn how to find the inverse of this part.  Only for now, we want to rely our application of the concept of inverse functions, not a procedure.

Last Example:

1. Determine if the relation is a function. Explain your reasoning.
2. Notice the domain and range of the graph.

Number i

You may have a piddling difficulty determining if this is a function or not.  To arroyo the question, first with what yous know.  What is x, hither?  Do you come across that it's an exponent?  The base is two.

Is at that place an exponent, for a base of ii, that has two answers?  All of the integers are fine.  two^three = 8, and ii^-2 = ¼.  Just what nearly the case where x = ½?  Remember than exponent of ½ is the same every bit square root.  The square root of two has a positive and negative value!  Therefore, this is Non a function.

Strangely enough, if you graphed this part with graphing software, it would pass the vertical line test.  The reason is that graphing calculators typically limit outputs (y values) so the graphs stay swell and easy to understand.

Number ii

The domain is all Real Numbers.  You can place any value you want for ten and raise two to that power.  The range is subtle.  While nosotros know the graph doesn't show all outputs for the equation y = 2 ¹⁰ ,

The y values appear to approaching the 10 axis, which is y = 0.  Merely, the output for y = 2 ¹⁰, will never exist goose egg.  2 to the power of negative 3 is 1/8.  Ii to the power of -5 is i/32.  The outputs arroyo cypher, just never impact information technology.  This is called an asymptote.

Notes on the vertical line examination:  The reason the vertical line test works is that it shows if in that location's an input with more than one output.  If a single input, mapped on a coordinate airplane, has more than one output, those 2 coordinates will align vertically.

Source: https://onteachingmath.com/courses/algebra1/functions/functions-day-2/

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