Problem 1:
If x is a real number, prove that the rational function
can have all numerical values except such as lie between 2 and 6. In other words, find the range of this rational function. (the domain of this rational function is all real numbers except
quite obviously.
Problem 2:
For all real values of x, prove that the quadratic function
has the same sign as a, except when the roots of the quadratic equation
are real and unequal, and x has a value lying between them. This is a very useful famous classic result.
Remarks:
a) From your proof, you can conclude the following also: The expression
will always have the same sign, whatever real value x may have, provided that
is negative or zero; and if this condition is satisfied, the expression is positive, or negative accordingly as a is positive or negative.
b) From your proof, and using the above conclusion, you can also conclude the following: Conversely, in order that the expression
may be always positive,
must be negative or zero; and, a must be positive; and, in order that
may be always negative,
must be negative or zero, and a must be negative.
Further Remarks:
Please note that the function
, where
and
is a parabola. The roots of this
are the points where the parabola cuts the y axis. Can you find the vertex of this parabola? Compare the graph of the elementary parabola
, with the graph of
where
and further with the graph of the general parabola
. Note you will just have to convert the expression
to a perfect square form.
Problem 3:
Find the limits between which a must lie in order that the rational function
may be real, if x is real.
Problem 4:
Determine the limits between which n must lie in order that the equation
may have real roots.
Problem 5:
If x be real, prove that
must lie between 1 and
.
Problem 6:
Prove that the range of the rational function
lies between 3 and
for all real values of x.
Problem 7:
If
, Prove that the rational function
can have no value between 5 and 9. In other words, prove that the range of the function is
.
Problem 8:
Find the equation whose roots are
.
Problem 9:
If
are roots of the quadratic equation
, find the value of (a)
(b)
.
Problem 10:
If the roots of
be in the ratio p:q, prove that 
Problem 11:
If x be real, the expression
admits of all values except such as those that lie between 2n and 2m.
Problem 12:
If the roots of the equation
are
and
, and those of the equation
be
and
, prove that
.
Problem 13:
Prove that the rational function
will be capable of all values when x is real, provided that p has any real value between 1 and 7. That is, under the conditions on p, we have to show that the given rational function has as its range the full real numbers. (Of course, the domain is real except those values of x for which the denominator is zero).
Problem 14:
Find the greatest value of
for any real value of x. (Remarks: this is maxima-minima problem which can be solved with algebra only, calculus is not needed).
Problem 15:
Show that if x is real, the expression
has no real value between b and a.
Problem 16:
If the roots of
be possible (real) and different, then the roots of
will not be real, and vice-versa. Prove this.
Problem 17:
Prove that the rational function
will be capable of all real values when x is real, if
and
have the same sign.
Cheers,
Nalin Pithwa