Exploring Holes and Discontinuities in Rational Functions

What defines a discontinuity in a rational function?

• Values where the function has a slope of 1

• Values where the function intersects the x-axis

• Values where the function is undefined

• Values where the function reaches its maximum

What is a hole in the context of rational functions?

• A point where the function has a maximum value

• A point where the function has a vertical asymptote

• A point where the function is undefined but can be defined by filling a single point

• A point where the function touches the x-axis

How do you find the x-coordinate of a hole in a rational function?

• By setting the numerator equal to zero

• By setting the denominator equal to zero

• By finding the derivative of the function

• By canceling out common factors in the numerator and denominator

What is the y-coordinate of the hole in the function f(x) = (x+2)/(x^2-x-6) after simplification?

• 2

• -2

• 0.2

• -0.2

How is a vertical asymptote determined in a rational function?

• By identifying values that make the simplified fraction undefined

• By setting the numerator equal to zero

• By identifying the maximum value of the function

• By finding the derivative of the function

In the second example, what causes a hole in the function?

• The derivative of the function

• The highest degree of the polynomial

• Setting the numerator equal to zero

• A common factor in the numerator and denominator

What are the coordinates of the hole in the second example?

• (5, 0.5)

• (5, -0.5)

• (-5, 0.5)

• (-5, -0.5)

What indicates the absence of holes in a rational function?

• When the function has a vertical asymptote

• When the function intersects the x-axis

• When the function's derivative is zero

• When there are no common factors between the numerator and denominator

How are vertical asymptotes identified in the absence of holes?

• By setting the denominator equal to zero

• By setting the numerator equal to zero

• By finding the derivative of the function

• By canceling out common factors

What is the purpose of graphing the function after identifying holes and asymptotes?

• To find the x-intercepts of the function

• To calculate the derivative of the function

• To visualize the discontinuities and behavior of the function

• To determine the function's maximum and minimum values