SOLUTION: 6. (a) Find the domain of the function f(x) = {x / sqrt (5−3x) + x sqrt (x + 1)} (b) Consider f(x) = sqrt (4x^2 + 1 ) and g(x) = x + 3/ x^2 . Find the composite functions

Algebra ->  Functions -> SOLUTION: 6. (a) Find the domain of the function f(x) = {x / sqrt (5−3x) + x sqrt (x + 1)} (b) Consider f(x) = sqrt (4x^2 + 1 ) and g(x) = x + 3/ x^2 . Find the composite functions      Log On


   

Question 1026181: 6. (a) Find the domain of the function
f(x) = {x / sqrt (5−3x) + x sqrt (x + 1)}
(b) Consider f(x) = sqrt (4x^2 + 1 ) and g(x) = x + 3/ x^2 . Find the composite functions f ◦g and g◦f.
(c) Consider h(x) = x^3 + 5 and k(x) = x^4 + 2x + 3. Find function l such that h◦l = k.
Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
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6. (a) Find the domain of the function
f(x) = {x / sqrt (5-3x) + x sqrt (x + 1)}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

When you see a square root, and you are asked about the domain, take care that the function under the square root is non-negative.

So, in this case these two inequalities must be held:

5-3x > 0,    (1)   and
x+1   = 0.   (2).

Notice that in (1) I wrote the strong inequality.
It is because we have sqrt%285-3x%29 in the denominator.


Now, from (1),  5 > 3x,  x < 5%2F3.

From (2),  x >= -1.


So, the answer is -1 <= x < 5%2F3.


It is the semi-segment  [-1, 5%2F3).


The domain is the semi-segment  [-1, 5%2F3).