By allowing for quotients and fractional powers in polynomial functions, we create a larger class of functions. An
algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots. Two types of algebraic functions are rational functions and root functions.
Just as rational numbers are quotients of integers, rational functions are quotients of polynomials. In particular, a
rational function is any function of the form
where
and
are polynomials. For example,
are rational functions. A
root function is a power function of the form
where
is a positive integer greater than one. For example,
is the square-root function and
is the cube-root function. By allowing for compositions of root functions and rational functions, we can create other algebraic functions. For example,
is an algebraic function.
Finding domain and range for algebraic functions
For each of the following functions, find the domain and range.
It is not possible to divide by zero, so the domain is the set of real numbers
such that
To find the range, we need to find the values
for which there exists a real number
such that
When we multiply both sides of this equation by
we see that
must satisfy the equation
From this equation, we can see that
must satisfy
If
this equation has no solution. On the other hand, as long as
satisfies this equation. We can conclude that the range of
is
To find the domain of
we need
When we factor, we write
This inequality holds if and only if both terms are positive or both terms are negative. For both terms to be positive, we need to find
such that
These two inequalities reduce to
and
Therefore, the set
must be part of the domain. For both terms to be negative, we need
These two inequalities also reduce to
and
There are no values of
that satisfy both of these inequalities. Thus, we can conclude the domain of this function is
If
then
Therefore,
and the range of
is
The root functions
have defining characteristics depending on whether
is odd or even. For all even integers
the domain of
is the interval
For all odd integers
the domain of
is the set of all real numbers. Since
for odd integers
is an odd function if
is odd. See the graphs of root functions for different values of
in
[link] .
Finding domains for algebraic functions
For each of the following functions, determine the domain of the function.
You cannot divide by zero, so the domain is the set of values
such that
Therefore, the domain is
You need to determine the values of
for which the denominator is zero. Since
for all real numbers
the denominator is never zero. Therefore, the domain is
Since the square root of a negative number is not a real number, the domain is the set of values
for which
Therefore, the domain is
The cube root is defined for all real numbers, so the domain is the interval
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost
6% more than previous year? Provide with reasoning.
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Good day
How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200.
Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho
ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9
no Mpc is greater than 1
Y=100+.9Y+50
Y-.9Y=150
0.1Y/0.1=150/0.1
Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question
If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output?
discuss your answer using appropriate graph.