Finding the interval of expression having two quadratic equations.

[Sumedh]

Homework Statement

What will be the values of 'm' so that the range of the equation
$$y= \frac{mx^2+3x-4}{-4x^2+3x+m}$$

will be all real values i.e. $$y\epsilon (-\infty,\infty)$$given:x can take all real values.
any help or hint will be appreciated.

Homework Equations

The Attempt at a Solution

i tried to find the range of the numerator and the denominator
by using the formula
if a<0 then range= (-infinity, -D/4a)

if a>0 then range= (-D/4a, infinity)
i couldn't proceed further.
please help
or provide hints.

 

[Dick]

Homework Statement

What will be the values of 'm' so that the range of the equation
$$y= \frac{mx^2+3x-4}{-4x^2+3x+m}$$

will be all real values( i.e. $$y\epsilon [-\infty,\infty]$$).


given:x can take all real values.
any help or hint will be appreciated.

Homework Equations

The Attempt at a Solution

i tried to find the range of the numerator and the denominator
by using the formula
if a<0 then range= (-infinity, -D/4a)

if a>0 then range= (-D/4a, infinity)
i couldn't proceed further.
please help
or provide hints.

I have no idea what 'a' or 'D' are supposed to be. Here's a hint. What happens if the denominator is never equal to zero? For what values of m is the denominator never equal to zero?

 

[Sumedh]

D= Discriminant
a=coefficient of x^2

 

[Dick]

D= Discriminant
a=coefficient of x^2

Well ok, that's fine then. Again, think about what happens if the denominator is never equal to zero. That means the denominator is always the same sign. Can the range be [-infinity,infinity]? If so how?

 

[Sumedh]

sorry the range should be(-infinity, infinity)
i corrected the brackets [] -->()



if the denominator is of same sign
then to get all real values from the equation
only the numerator will have to be considered

 

[Dick]

sorry the range should be(-infinity, infinity)
i corrected the brackets [] -->()



if the denominator is of same sign
then to get all real values from the equation
only the numerator will have to be considered

Yes, so consider the numerator. If it has a lower or upper bound can the range be (-infinity,infinity)?

 

[Sumedh]

If it has a lower or upper bound can the range be (-infinity,infinity)?

lower or upper bound =?if denominator is not zero and coefficient of x^2 is negative then the whole denominator will be negative

now as the denominator is negative
if numerator is positive the whole equation will be negative
and
if numerator is negative the whole equation will be positive

 

[Dick]

lower or upper bound =?


if denominator is not zero and coefficient of x^2 is negative then the whole denominator will be negative

now as the denominator is negative
if numerator is positive the whole equation will be negative
and
if numerator is negative the whole equation will be positive

Try a concrete example. Suppose the range of the denominator is (-inf,-1/2] (so it's never zero) and the range of the numerator is [-1,inf). Can you show in that case that the range of the ratio isn't (-inf,inf)? Can you find a number that can't be in the range?

 

[eumyang]

OP has already posted this question here:
https://www.physicsforums.com/showthread.php?t=523904"

 

[Sumedh]

Can you find a number that can't be in the range?

no

thank you very much.