Inverse Functions - Algebra 2

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

One definition of the inverse function of a function is the set of all ordered pairs such that the ordered pair is in the set of ordered pairs in . As such, if the ordered pairs of are given, as is the case here, the set of ordered pairs in can be found by switching the positions of the coordinates in all of the pairs. Doing this, we obtain:

or, ordering the -coordinates,

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to be the case:

If follows that has an inverse function.

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

One definition of the inverse function of a function is the set of all ordered pairs such that the ordered pair is in the set of ordered pairs in . As such, if the ordered pairs of are given, as is the case here, the set of ordered pairs in can be found by switching the positions of the coordinates in all of the pairs. Doing this, we obtain:

or, ordering the -coordinates,

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to be the case:

If follows that has an inverse function.

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

One definition of the inverse function of a function is the set of all ordered pairs such that the ordered pair is in the set of ordered pairs in . As such, if the ordered pairs of are given, as is the case here, the set of ordered pairs in can be found by switching the positions of the coordinates in all of the pairs. Doing this, we obtain:

or, ordering the -coordinates,

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to be the case:

If follows that has an inverse function.

A function has an inverse function if and only if, for all in the domain of , if , it follows that . In other words, no two values in the domain can be matched with the same range value.

If we order the rows by range value, we see this to not be the case:

and . Since two range values exist to which more than one domain value is matched, the function has no inverse.

One definition of the inverse function of a function is the set of all ordered pairs such that the ordered pair is in the set of ordered pairs in . As such, if the ordered pairs of are given, as is the case here, the set of ordered pairs in can be found by switching the positions of the coordinates in all of the pairs. Doing this, we obtain:

or, ordering the -coordinates,

To create the inverse, switch x and y making the solution x=3y+3.

y must be isolated to finish the problem.

Interchange the x and y-variables.

Solve for y. Distribute the constant through the binomial.

Add 24 on both sides.

The equation becomes:

Divide by four on both sides.

Simplify both sides.

The answer is:

Given

Hence

Interchanging with we get:

Solving for results in .

The question is asking for the inverse function. To find the inverse, first switch input and output -- which is usually easiest if you use notation instead of . Then, solve for .

Here's where we switch:

To solve for , we first have to get it out of the denominator. We do that by multiplying both sides by .

Distribute:

Get all the terms on the same side of the equation:

Factor out a :

Divide by :

This is our inverse function!

In order to find the inverse function, we must swap and and then solve for .

Becomes

Now we need to solve for :

Finally, we need to divide each side by 4.

This gives us our inverse function:

Let's say that the function takes the input and yields the output . In math terms:

So, the inverse function needs to take the input and yield the output :

So, to answer this question, we need to flip the inputs and outputs for . We do this by replacing with (or a dummy variable; I used ) and with . Then we solve for to get our inverse function:

Now we solve for by subtracting from both sides, taking the cube root, and then adding :

is our inverse function,

The question is essentially asking this: take say that equals , then take , then whatever that equals, say , take . So, we start with ; we know that , so if we flip that around we know . Now we have to take , but we know that is . Now we have to take , but we don't have that in our table; we do have , though, and if we flip it around, we get , which is our answer.

Our question is asking "What is of of inverse?" First we find the inverse of . Looking at the question, we see ; if we flip that around, we get . Now we need to find what is; that is an easy one, as it is directly provided: . Now we need to find . Again, this isn't given, but what is given is , so , and that is our answer.