Question

The "panvoid" of a function is defined as the sum of the integers that do not fall within the domain of the function. All of the following functions have equal panvoids EXCEPT

• A. $f\left(x\right)=\frac{3x}{x^2-4}$
• B. $f\left(x\right)=\frac{3-x}{x^3-x}$
• C. $f\left(x\right)=\frac{3x}{3x^2-27}$
• D. $f\left(x\right)=\frac{x+2}{x}$
• E. $f\left(x\right)=\frac{2-x}{x^2-x}$

Answer 1

The correct option is E $f\left(x\right)=\frac{2-x}{x^2-x}$

A. $f\left(x\right)=\frac{3x}{x^2-4}$

Since the denominator of the function can never be equal to zero for it to yield a valid answer, $x^2-4\ne 0$

$\Rightarrow x\ne \pm 2$

Panvoid $=2-2=0$

B. $f\left(x\right)=\frac{3-x}{x^3-x}$

Since the denominator of the function can never be equal to zero for it to yield a valid answer, $x^3-x\ne 0$

Sum of roots of this cubic equation is zero.

Hence panvoid $=0$

C. $f\left(x\right)=\frac{3x}{3x^2-27}$

Since the denominator of the function can never be equal to zero for it to yield a valid answer, $3x^2-27\ne 0$

$\Rightarrow x^2-9\ne 0$

$\Rightarrow x\ne \pm 3$

Panvoid $=3-3=0$

D. $f\left(x\right)=\frac{x+2}{x}$

Since the denominator of the function can never be equal to zero for it to yield a valid answer, $x\ne 0$

Panvoid $=0$

E. $f\left(x\right)=\frac{2-x}{x^2-x}$

Since the denominator of the function can never be equal to zero for it to yield a valid answer, $x^2-x\ne 0$

$\Rightarrow x(x-1)\ne 0$

$\Rightarrow x\ne 0,1$

Panvoid $=1$