identify any discontinuities in the function g(x)=(x^3 - 9x)/(x^2 - 6x + 9). a) there are two…

identify any discontinuities in the function g(x)=(x^3 - 9x)/(x^2 - 6x + 9). a) there are two discontinuities at x = 0 and x = 3. b) there is one discontinuity at x = 3. c) there are no discontinuities. d) there are two discontinuities at x = 2 and x = 3.

identify any discontinuities in the function g(x)=(x^3 - 9x)/(x^2 - 6x + 9). a) there are two discontinuities at x = 0 and x = 3. b) there is one discontinuity at x = 3. c) there are no discontinuities. d) there are two discontinuities at x = 2 and x = 3.

Answer

Explanation:

Step1: Factor the function

First, factor the numerator and denominator. The numerator $x^{3}-9x=x(x^{2} - 9)=x(x - 3)(x + 3)$. The denominator $x^{2}-6x + 9=(x - 3)^{2}$. So, $g(x)=\frac{x(x - 3)(x + 3)}{(x - 3)^{2}}=\frac{x(x + 3)}{x - 3},x\neq3$.

Step2: Find the discontinuities

A rational - function $y=\frac{f(x)}{h(x)}$ is discontinuous at the values of $x$ for which $h(x)=0$. Set the denominator $x - 3=0$, we get $x = 3$. When $x = 3$, the function $g(x)$ is undefined.

Answer:

B. There is one discontinuity at $x = 3$.