Understanding Rational Functions

The video tutorial explains how to analyze a rational function by determining its domain, identifying intervals where the function is increasing or decreasing, and assessing its concavity. The process involves setting the denominator to zero to find the domain, using the first derivative to find critical numbers and test intervals for increasing or decreasing behavior, and applying the second derivative to determine concavity. The tutorial also includes graphical verification of these properties.

What is the first step in analyzing a rational function?

• Finding the range

• Determining the domain

• Calculating the second derivative

• Identifying asymptotes

How do you find the domain of a rational function?

• By finding the y-intercepts

• By finding the x-intercepts

• By setting the numerator equal to zero

• By setting the denominator equal to zero

What is the domain of the function if the denominator is 3x + 1?

• All real numbers except x = 1/3

• All real numbers

• All real numbers except x = 0

• All real numbers except x = -1/3

What indicates that a function is increasing on an interval?

• The second derivative is positive

• The first derivative is negative

• The first derivative is positive

• The first derivative is zero

What is the behavior of the function over its entire domain?

• Constant

• Decreasing

• Increasing

• Oscillating

What does a vertical asymptote at x = -1/3 indicate?

• The function is undefined at x = -1/3

• The function is zero at x = -1/3

• The function is maximum at x = -1/3

• The function is minimum at x = -1/3

What does it mean if the second derivative is positive?

• The function is concave down

• The function is concave up

• The function is decreasing

• The function is increasing

How do you determine the concavity of a function?

• By finding the third derivative

• By finding the critical points

• By finding the second derivative

• By finding the first derivative

What is the concavity of the function on the interval from -1/3 to infinity?

• Neither

• Both

• Concave down

• Concave up

What shape does a concave up function resemble?

• A frown

• A smile

• A straight line

• A zigzag