Here are the problems I need help with:

Take the following rational function and express it as the sum of a series of rational functions whose numerators do not exceed degree 1 and whose respective denominators exceed their numerators by degree one.

() = (x^2+3x+2)/(x^3-ax^2-x+a)

find the domain and range of the following equation

y=log*2*(tan(sqrtx))
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1 Answer

(1) The denominator can be factorised:

x³-ax²-x+a=x³-x-(ax²-a)=x(x²-1)-a(x²-1)=(x-a)(x²-1)=(x-a)(x-1)(x+1).

The numerator can be factorised:

(x+2)(x+1).

The common factor x+1 can be cancelled (for x≠-1) so the expression is (x+2)/((x-1)(x-a).

We can use partial fractions:

(x+2)/((x-1)(x-a))≡A/(x-1)+B/(x-a) where A and B are constants:

x+2≡A(x-a)+B(x-1)=Ax-Aa+Bx-B.

Matching coefficients:

A+B=1, -Aa-B=2, add these equations: A-Aa=3, A(1-a)=3, A=3/(1-a).

B=1-A=1-3/(1-a)=(1-a-3)/(1-a)=-(2+a)/(1-a)

So the expression becomes 3/((x-1)(1-a))-(2+a)/((x-a)(1-a)).

(2) I read this as y=log₂(tan(√x)).

The argument of square root must be positive, and the argument of log must be strictly positive. The argument of tangent must be less than π/2. Therefore 0<x<π²/4, and the range is (-∞,∞). However, √x can be in the third quadrant where tangent is positive. π<√x<3π/2, meaning π²<x<9π²/4. This pattern repeats every cycle, without altering the range, but the domain is discontinuous. There is no valid value of x between π²/4 and π². Each domain region gets progressively wider.

by Top Rated User (815k points)

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