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Functions
[Show All Solutions]
1. Given 
and find each of the following.
(a) (b) (c) (d)
(e) (f) (g) (h)
Show Solution
All throughout a calculus sequence
you will be asked to deal with functions so make sure that you are familiar and
comfortable with the notation and can evaluate functions.
First recall that the in a function is nothing more than a fancy way
of writing the y in an equation so
is equivalent to writing
except the function notation form,
while messier to write, is much more convenient for the types of problem you’ll
be working in a Calculus class.
In this problem we’re asked to
evaluate some functions. So, in the
first case is asking us to determine the value of when .
The key to remembering how to
evaluate functions is to remember that you whatever is in the parenthesis on
the left is substituted in for all the x’s
on the right side.
So, here are the function
evaluations.
(a)
(b)
(c)
(d)
(e)
Remember that we substitute for the
x’s WHATEVER is in the parenthesis on
the left. Often this will be something
other than a number. So, in this case we
put t’s in for all the x’s on the left.
This is the same as we did for (a) (d) except we are now substituting in
something other than a number.
Evaluation works the same regardless of whether we are substituting a
number or something more complicated.
(f)
Often instead of evaluating
functions at numbers or single letters we will have some fairly complex
evaluations so make sure that you can do these kinds of evaluations.
(g)
The only difference between this
one and the previous one is that I changed the t to an x. Other than that there is absolutely no
difference between the two!
Do not let the fact that there are x’s in the parenthesis on the left get
you worked up! Simply replace all the x’s in the formula on the right side
with . This one works exactly the same as(f).
(h)
Do not get excited by problems
like (e) (h). This type of problem works the same
as (a) (d) we just aren’t using numbers! Instead of substituting numbers you are
substituting letters and/or other functions.
So, if you can do (a) (d)
you can do these more complex function evaluations as well!
2. Given find each of the following.
(a) (b) (c)
Show Solution
This is one of the simplest functions in the world to evaluate, but for some
reason seems to cause no end of difficulty for students. Recall from the previous problem how function
evaluation works. We replace every x on the right side with what ever is in
the parenthesis on the left. However, in
this case since there are no x’s on
the right side (this is probably what causes the problems) we simply get 10 out
of each of the function evaluations.
This kind of function is called a constant
function. Just to be clear here are
the function evaluations.
3. Given and find each of the following.
(a) (b) (c)
(d) (e) (d)
Show Solution
This problem makes sure you are
familiar with notation commonly used with functions. The appropriate formulas are included in the
answer to each part.
(a)
(b)
(c)
(d) For this problems (and the next two) remember
that the little circle , ,
in this problem signifies that we are doing composition NOT multiplication!
The basic formula for composition
is
In other words, you plug the second
function listed into the first function listed then evaluate as appropriate.
In this case we’ve got a number
instead of an x but it works in
exactly the same way.
(e) Compare the results of
this problem to (c)! Composition is NOT the same as multiplication
so be careful to not confuse the two!
(f) Compare the results of this to (e)!
The order in which composition is written is important! Make sure you pay attention to the order.