which statement describes the domain of the function $f(x)=\frac{3x}{4x^{2}-4}$?\nall real numbers\nall…

which statement describes the domain of the function $f(x)=\frac{3x}{4x^{2}-4}$?\nall real numbers\nall nonzero real numbers\nall real numbers except $x = \frac{3}{4}$\nall real numbers except $x=-1$ and $x = 1$

which statement describes the domain of the function $f(x)=\frac{3x}{4x^{2}-4}$?\nall real numbers\nall nonzero real numbers\nall real numbers except $x = \frac{3}{4}$\nall real numbers except $x=-1$ and $x = 1$

Answer

Explanation:

Step1: Recall domain - denominator non - zero

The domain of a rational function is all real numbers except where the denominator is zero. Set the denominator equal to zero: $4x^{2}-4 = 0$.

Step2: Solve the equation for x

Factor out a 4 from the left - hand side: $4(x^{2}-1)=0$. Then, since $x^{2}-1=(x + 1)(x - 1)$ by the difference of squares formula, we have $4(x + 1)(x - 1)=0$. Divide both sides by 4 to get $(x + 1)(x - 1)=0$. Using the zero - product property, if $ab = 0$, then $a = 0$ or $b = 0$. So $x+1 = 0$ gives $x=-1$ and $x - 1=0$ gives $x = 1$.

Answer:

D. all real numbers except $x=-1$ and $x = 1$