The domain of is only .
horizontal
Composition with a linear function as the inside function results in horizontal graphical transformations.
Briefly: moves along the real line left to right from to . We first encounter an
included endpoint and then a short line segment, then a corner at , then a longer line
segment with an excluded endpoint.
And, now compose it with
We have three functions: , , and . We would like to keep these separate in our thoughts. Therefore, we will have three separate variables representing their dmomains.
The implied, natural domain for is . The natural range for is . This does not match the domain of .
Therefore, we must align those.
But, before doing that, let’s just think about traversing the real number line.
We can imagine moving along the real line left to right from to . These numbers are input into and the output runs right to left from to . The direction is reversed because of the coefficient in . It is changing the sign of the numbers.
Therefore, in the composition, the domain of is running backwards. From right to left, the composition will first encounter the longer line segment with the excluded endpoint, then a corner, then the shorter line segment with the included endpoint.
The graph is reflected horizontally.
That was due to the negative sign of the coefficient. Now to figure out the effect of , which is less than .
The domain of is . So, when do the values of equal and in the range?
Since all of the domain numbers of are multiplied by , these function values, and , of
occur at and in the domain of .
The domain of is .
will move across the interval , from left to right.
These domain values will be multiplied by and the function values of will move across the interval from right to left. These values are going into backwards from their normal order.
The graph of will first encounter the longer line segment with the excluded endpoint, then it will cross the corner.
Then the graph of encounters the shorter line segment with the included endpoint.
Let with domain .
The graph has transformed horizontally.
- The negative coefficient in has reflected the graph horizontally.
- The coefficient means the domain needs to expand by a factor of , so that the values of match the normal inputs to .
The corner of the graph of an absolute value function occurs when the inside of the absolute value signs equal .
For , this means , which occurs when .
The corner on the graph is
Let with its natural or implied domain.
Feature Inventory:
- Domain is .
- Starting point is .
- Graph opens to the right.
- is an increasing function.
Since, is an increasing function, there should be one zero.
Now a Composition
Let with its natural or implied domain.
Define with its natural or implied domain.
will be taking real numbers and doing this to them: . AFTER it does that, the values will go into and at this time they need to match up to our inventory list.
Therefore, to locate all of the corresponding places in the domain of we must undo .
- add
- multiply by
The starting point is now .
The zero is now at
We also know that the graphs of square root functions begin when the inside equals .
Beginning at or .
has a zero, for which we can just solve.
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more examples can be found by following this link
More Examples of Transforming the Inside