determine if the given function has any points of discontinuity. explain your reasoning. f(x) = (x² - b²)/(x…

determine if the given function has any points of discontinuity. explain your reasoning. f(x) = (x² - b²)/(x - b) there is a point of discontinuity at x = b because the denominator has the factor x - b. there are points of discontinuity at both x = -b and x = b because the numerator has factors of x + b and x - b. there is a point of discontinuity at x = -b only because the factor of x - b is common to both the numerator and denominator. there is a point of discontinuity at x = b only because the factor of x - b is common to both the numerator and denominator and factors out.

determine if the given function has any points of discontinuity. explain your reasoning. f(x) = (x² - b²)/(x - b) there is a point of discontinuity at x = b because the denominator has the factor x - b. there are points of discontinuity at both x = -b and x = b because the numerator has factors of x + b and x - b. there is a point of discontinuity at x = -b only because the factor of x - b is common to both the numerator and denominator. there is a point of discontinuity at x = b only because the factor of x - b is common to both the numerator and denominator and factors out.

Answer

Answer:

A. There is a point of discontinuity at $x = b$ because the denominator has the factor $x - b$.

Explanation:

Step1: Recall discontinuity condition

A rational - function $y=\frac{f(x)}{g(x)}$ is discontinuous where $g(x)=0$.

Step2: Identify the denominator

For $f(x)=\frac{x^{2}-b^{2}}{x - b}$, the denominator is $g(x)=x - b$.

Step3: Find the zero of the denominator

Set $x - b=0$, solving for $x$ gives $x = b$. So the function is discontinuous at $x = b$. The numerator $x^{2}-b^{2}=(x + b)(x - b)$, but the discontinuity is determined by the zero of the original denominator before any simplification.