When we compose functions, we take a function of a function. For example, suppose the temperature
on a given day is described as a function of time
(measured in hours after midnight) as in
[link] . Suppose the cost
to heat or cool a building for 1 hour, can be described as a function of the temperature
Combining these two functions, we can describe the cost of heating or cooling a building as a function of time by evaluating
We have defined a new function, denoted
which is defined such that
for all
in the domain of
This new function is called a composite function. We note that since cost is a function of temperature and temperature is a function of time, it makes sense to define this new function
It does not make sense to consider
because temperature is not a function of cost.
Definition
Consider the function
with domain
and range
and the function
with domain
and range
If
is a subset of
then the
composite function
is the function with domain
such that
A composite function
can be viewed in two steps. First, the function
maps each input
in the domain of
to its output
in the range of
Second, since the range of
is a subset of the domain of
the output
is an element in the domain of
and therefore it is mapped to an output
in the range of
In
[link] , we see a visual image of a composite function.
Compositions of functions defined by formulas
Consider the functions
and
Find
and state its domain and range.
Evaluate
Find
and state its domain and range.
Evaluate
We can find the formula for
in two different ways. We could write
Alternatively, we could write
Since
for all real numbers
the domain of
is the set of all real numbers. Since
the range is, at most, the interval
To show that the range is this entire interval, we let
and solve this equation for
to show that for all
in the interval
there exists a real number
such that
Solving this equation for
we see that
which implies that
If
is in the interval
the expression under the radical is nonnegative, and therefore there exists a real number
such that
We conclude that the range of
is the interval
We can find a formula for
in two ways. First, we could write
Alternatively, we could write
The domain of
is the set of all real numbers
such that
To find the range of
we need to find all values
for which there exists a real number
such that
Solving this equation for
we see that we need
to satisfy
which simplifies to
Finally, we obtain
Since
is a real number if and only if
the range of
is the set
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities.
before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue..
sometimes due to these physical problems..the diagnosis may be misdiagnosed.
treatment for autism.
well it depends on the severity.
since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision.
play therapy.
speech therapy.
Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game.
string and beads game.
the more you interact with the child the more effective
Jharna
results you'll get..
please consult a therapist to know what suits best on your child.
and last as a parent. I know sometimes it's overwhelming to guide a special kid.
but trust the process and be strong and patient as a parent.
Jharna
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