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what is the range of (fxg):
f(x)=2x^2+3
g(x)=1/x
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The function (fxg)(x) is F(x) = = 2x + .
They ask about the range of this function.
The plot is shown in this link
https://www.desmos.com/calculator/lh7qjcdhic
https://www.desmos.com/calculator/lh7qjcdhic
There are two ways to solve this problem: one way is Algebra, and another way is Calculus.
Algebra way
The range is the set of real values t such that
2x + = t for some x.
In other words, the range is the set of real numbers t such that the quadratic equation
2x^2 + 3 = tx, or 2x^2 - tx + 3 = 0
has real solutions for x.
For it, the necessary and sufficient condition is that the discriminant is non-negative
d = (-t)^2 - 4*2*3 >= 0, or t^2 >= 24, or |t| >= , or
+------------------------------------+
| t <= or t >= . |
+------------------------------------+
Thus the range of the function (fxg) is the union of the two sets
(,] U [,). ANSWER