Given #y={2x^2-a}/{x^2-4}# Find: i) the domain ii) the range iii) x and y intercepts iv) the asymptotes v) point of inflexion?
Given #y={2x^2-a}/(x^2-4)#
Find: i) the domain
ii) the range
iii) x and y intercepts
iv) the asymptotes
v) point of inflection
Given
Find: i) the domain
ii) the range
iii) x and y intercepts
iv) the asymptotes
v) point of inflection
2 Answers
Some confusion about the function, let's go with
We're working over the reals presumably so the domain is all the reals except where the denominator vanishes,
So the domain is
The range is tricky. Let's dispense with special case
The range when
When
By the symmetry the stationary point must be
We're already assuming
As
The range is
We write the funny conjunction to handle both
We just calculated the y intercept as
The x intercept aka zero is when the numerator is zero,
We calculated the inflection point as the y intercept,
I've had the graph crash my tab so I'll post first then add the graph.
Let's graph the cases
Please see the explanation below.
Explanation:
The function is
The denominator must be
The domain is
Calculate the range as follows
Therefore,
and
The range is
The intercept are as follows:
y axis when
The point of intercept with the y-axis is
x axis when
The point of intercept with the x-axis is
The vertical asymptotes are at
The horizontal asymptotes are as follows
The horizontal asymptote is
There are no slant asymptotes as the degree of the numerator is
Calculate the first and second derivatives.
As
Therefore,
So, there are no points of inflections
Graph for
graph{(2x^2-5)/(x^2-4) [-10, 10, -5, 5]}