.which range of the function between f and g is equal to the range of the function f.g(x)?
please help me with my exam is going on please donot write wrong answer beacause it is my exam
Answers
Step-by-step explanation:
Composition of
functions
mc-TY-composite-2009-1
We can build up complicated functions from simple functions by using the process of composition,
where the output of one function becomes the input of another. It is also sometimes necessary
to carry out the reverse process, decomposing a complicated function into two or more simple
functions.
In order to master the techniques explained here it is vital that you undertake plenty of practice
exercises so that they become second nature.
After reading this text, and/or viewing the video tutorial on this topic, you should be able to:
• write down both the composite functions gf and fg given two suitable functions f and g,
• write a complicated function as a composition gf,
• determine whether two given functions f and g are suitable for composition,
• find the domain and range of a composite function gf given the functions f and g.
Contents
1. Introduction 2
2. Order of composition 3
3. Decomposition of a function 3
4. Domains and ranges of composed functions 4
1. Introduction
The composition of two functions g and f is the new function we get by performing f first, and
then performing g. For example, if we let f be the function given by f(x) = x
2 and let g be the
function given by g(x) = x + 3, then the composition of g with f is called gf and is worked out
as
gf(x) = g(f(x)).
So we write down what f(x) is first, and then we apply g to the whole of f(x). In this case, if we
apply g to something we add 3 to it. So if we apply g to x
2
, we add three to x
2
. So we obtain
gf(x) = g(f(x)) = g(x
2
) = x
2 + 3 .
Here is another example of composition of functions. This time let f be the function given by
f(x) = 2x and let g be the function given by g(x) = e
x
. As before, we write down f(x) first, and
then apply g to the whole of f(x). In this case, f(x) is just 2x. Applying the function g then
raises e to the power f(x). So we obtain
gf(x) = g(f(x)) = g(2x) = e
2x
.
Sometimes the composition of two functions is called a ‘function of a function’, and sometimes
gf is written g◦f. You don’t have to use this notation yourself, but it is a good idea to remember
what it means because you might see it used in textbooks.
Key Point
A composed function gf is the function given by
gf(x) = g(f(x)).
This is sometimes called a function of a function. An alternative notation for gf is g ◦ f.
HOPE THIS IS HELPFUL