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Example Questions
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
Which of the following values ofxis not in the domain of the functiony= (2x –1) / (x2– 6x+ 9) ?
1/2
0
–1/2
2
3
3
Values ofxthat make the denominator equal zero are not included in the domain. The denominator can be simplified to (x –3)2, so the value that makes it zero is 3.
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
Given the relation below:
{(1, 2), (3, 4), (5, 6), (7, 8)}
Find the range of the inverse of the relation.
The domain of a relation is the same as the range of the inverse of the relation. In other words, the x-values of the relation are the y-values of the inverse.
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
What is the range of the functiony=x2+ 2?
y≥ 2
{2}
{–2, 2}
all real numbers
undefined
y≥ 2
The range of a function is the set ofy-values that a function can take. First let's find the domain. The domain is the set ofx-values that the function can take. Here the domain is all real numbers because nox-value will make this function undefined. (Dividing by 0 is an example of an operation that would make the function undefined.)
So if any value ofxcan be plugged intoy=x2+ 2,可以ytake any value also? Not quite! The smallest value thatycan ever be is 2. No matter what value ofxis plugged in,y=x2+ 2 will never produce a number less than 2. Therefore the range isy≥ 2.
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
What is the smallest value that belongs to the range of the function?
We need to be careful here not to confuse the domain and range of a function. The problem specifically concerns the range of the function, which is the set of possible numbers of. It can be helpful to think of the range as all the possible y-values we could have on the points on the graph of.
Notice thathasin its equation. Whenever we have an absolute value of some quantity, the result will always be equal to or greater than zero. In other words, |4-x|0. We are asked to find the smallest value in the range of, so let's consider the smallest value of, which would have to be zero. Let's see what would happen toif.
This means that when,. Let's see what happens whengets larger. For example, let's let.
As we can see, asgets larger, so does. We wantto be as small as possible, so we are going to wantto be equal to zero. And, as we already determiend,equalswhen.
The answer is.
Example Question #2 :How To Find Domain And Range Of The Inverse Of A Relation
我f, then find
is the same as.
To find the inverse simply exchangeandand solve for.
So we getwhich leads to.
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
我f, then which of the following is equal to?
Example Question #2 :How To Find Domain And Range Of The Inverse Of A Relation
Given the relation below, identify the domain of the inverse of the relation.
The inverse of the relation does not exist.
The domain of the inverse of a relation is the same as the range of the original relation. In other words, the y-values of the relation are the x-values of the inverse.
For the original relation, the range is:.
Thus, the domain for the inverse relation will also be.
Example Question #8 :How To Find Domain And Range Of The Inverse Of A Relation
Define, restricting the domain of the function to.
Determine(you need not determine its domain restriction).
does not exist
First, we must determine whetherexists.
A quadratic function has a parabola as its graph; this graph decreases, then increases (or vice versa), with a vertex at which the change takes place.
exists if and only if, if, then- or, equivalently, if there doesnotexistandsuch that, but. This will happen on any interval on which the graph ofconstantly increases or constantly decreases, but if the graph changes direction on an interval, there will besuch thaton this interval. The key is therefore to determine whether the interval to which the domain is restricted contains the vertex.
The-coordinate of the vertex of the parabola of the function
is.
The-coordinate of the vertex of the parabola ofcan be found by setting:
.
The vertex of the graph ofwithoutits domain restriction is at the point with-coordinate 2. However,. Therefore, the domain at whichis restricted does not include the vertex, andexists on this domain.
To determine the inverse of, first, rewritein vertex form
, the same asin the standard form.
The graph of, if unrestricted, would have a vertex with-coordinate 2, and-coordinate
.
Therefore,.
The vertex form ofis therefore
To find, first replacewith:
Switchand:
Solve for. First, add 8 to both sides:
Take the square root of both sides:
Add 2 to both sides
Replacewith:
Eitheror
The domain of是负的数字的集合;这是consequently the range of.can only have positive values, so the only possible choice foris.
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
Define a function.
我t is desired that is domain be restricted so thathave an inverse. Which of these domain restrictions wouldnotachieve that goal?
A quadratic function has a parabola as its graph; this graph decreases, then increases (or vice versa), with a vertex at which the change takes place.
exists if and only if, if, then- or, equivalently, if there doesnotexistandsuch that, but. This will happen on any interval on which the graph ofconstantly increases or constantly decreases, but if the graph changes direction on an interval, there will besuch thaton this interval. The key is therefore to identify the interval that contains the vertex.
The-coordinate of the vertex of the parabola of the function
is.
The-coordinate of the vertex of the parabola ofcan be found by setting:
.
Of the five intervals in the choices,
,
socannot exist ifis restricted to this interval. This is the correct choice.
Example Question #1 :How To Find Domain And Range Of The Inverse Of A Relation
Define, restricting the domain of the function to.
Determine(you need not determine its domain restriction).
does not exist
does not exist
First, we must determine whetherexists.
A quadratic function has a parabola as its graph; this graph changes direction (downward to upward, or vice versa) at a given point called the vertex.
exists on a given domain interval if and only if there doesnotexistandon this domain such that, but. This will happen if the graph changes direction on the domain interval. The key is therefore to determine whether the given domain interval includes the vertex.
The-coordinate of the vertex of the parabola of the function
is.
The-coordinate of the vertex of the parabola ofcan be found by setting:
.
The vertex of the graph ofwithoutits domain restriction is at the point with-coordinate 8. Since, the vertex is in the interior of the domain; as a consequence, the graph ofchanges direction on the interval, anddoes not exist on.
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