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.Â *This is a very useful famous classic result.Â *

*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).Â *

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