SOLUTION: Given the functions 𝑓(𝑥) = 2𝑥 − 1 , 𝑥 ∈ ]1; 10[ , 𝑔(𝑥) = 3𝑥 + 1 , 𝑥 ∈ ]−1; two[ . Calculate the domain and the correspondence rule of a) đ

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Given the functions 𝑓(𝑥) = 2𝑥 − 1 , 𝑥 ∈ ]1; 10[ , 𝑔(𝑥) = 3𝑥 + 1 , 𝑥 ∈ ]−1; two[ . Calculate the domain and the correspondence rule of a) đ      Log On


   

Question 1194817: Given the functions
𝑓(𝑥) = 2𝑥 − 1 , 𝑥 ∈ ]1; 10[ ,
𝑔(𝑥) = 3𝑥 + 1 , 𝑥 ∈ ]−1; two[ .
Calculate the domain
and the correspondence rule of
a) 𝑓 ∘ 𝑔
b) 𝑔 ∘ f
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
To find the equation for f∘g(x), plug the entire right side of the 
equation for g(x), for x in the equation for f(x).

f(x) = 2x-1,   g(x) = 3x+1

So we plug (3x+1) in place of x in 2x-1 and get 2(3x+1)-1 which 
simplifies to 6x+2-1 or 6x+1, so the equation for f∘g(x) is

f∘g(x) = 6x+1.

Now we must find the domain for f∘g(x). 

Any value we substitute for x in f∘g(x) must:
(a) be in the domain of g(x).
(b) g(x) must produce a value in the domain of f(x)

g(x) has the domain  ]-1; 2[ 

So the domain for f∘g(x) must be contained entirely within this 
interval.

Since g(x) is linear, the endpoints of the range of g(x) is found
by substituting the end points of the domain of g(x) for x in the
equation for g(x): 3(-1)+2=-1, 3(2)+2=8 so g(x) produces the range
]-1; 8[ 

But f(x) has the domain  ]1; 10[ , so f(x) cannot be used for 
]-1; 1],
so we must remove  ]-1; 1]  from ]-1; 2[ which leaves ]1; 2[ 

So:

f∘g(x) = 6x+1, x ∈ ]1; 2[ 

Now you switch the roles of f and g and do part b) yourself.

Edwin

Answer by ikleyn(52799) About Me  (Show Source):
You can put this solution on YOUR website!
.

Hello, I have lived in the world for many years and saw many different ways
to define the domain of a function, but never saw this way to write the domain as you use, like ]1;10[.


What does it mean this mysterious designation: open interval (1,10) with excluded endpoints or,
in opposite, everything OUTSIDE of this interval ?


And another question: why do you use non-standard notification, which is NEVER used in Math,
and, in addition, introduce it without any explanations ?


*********************************************************
        Doing this way is standardly considered in Math
                        as a Math crime.
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Usually, when the problem comes with non-standard and UNDEFINED designation, it ONLY deserves to be
returned to its author with a notification from a reader - - - it is a standard way to react on such parcels.


Writing accurately is a form of mathematical politeness.