How to find the domain and the range of a rational function?

Summary

This script explains the process of determining the domain and range of a function, focusing on a specific example. It emphasizes the importance of identifying restrictions, such as where the function is undefined, and how to factor and cancel terms to simplify the function. The script also discusses graphing the function, including the y-intercept and slope, and the significance of open circles on the graph to represent points where the function is not defined. Finally, it touches on the concept of limits to find missing y-values at restricted points.

Takeaways

• 🔍 The domain of the function is determined by setting the bottom of the fraction to zero, resulting in x ≠ 3.
• 📐 The domain is expressed as two intervals: (-∞, 3) U (3, ∞), excluding the point where x = 3.
• 🔢 Factoring and canceling terms in the function simplifies it to x + 1/4, indicating a linear relationship.
• ✏️ The function is not a straight line due to the restriction x ≠ 3, which must be considered in the graph.
• 📊 The graph of the function starts with the y-intercept at 1/4 and has a slope of 1/4, representing a line with a gentle incline.
• 🚫 At x = 3, the function is undefined, so an open circle is used on the graph to denote this exclusion.
• 📈 The y-value at the open circle (x = 3) is found by substituting x = 3 into the original function, resulting in y = 1.
• 📋 The range of the function is from negative infinity to just below 1, and then from 1 to infinity, with a hole at y = 1.
• 🔄 The concept of limits is introduced to understand the behavior of the function around the excluded value x = 3.
• 📝 The final graph of the function includes a hole at (3, 1), indicating that 3 is not in the domain and 1 is not in the range.

Q & A

What is the first step to determine the domain of the function described in the script?

The first step is to set the denominator of the function equal to zero and solve for x, ensuring that the denominator does not equal zero in the domain.

What is the restriction for the domain of the function based on the script?

The domain is restricted such that x cannot be equal to 3, as this would make the denominator zero.

How is the domain of the function described in the script?

The domain is described as all real numbers from negative infinity to 3, not including 3, and from 3 to infinity.

What is the process to find the range of the function as explained in the script?

The range is found by factoring and canceling terms in the numerator and denominator, resulting in a simplified function of (x + 1)/4.

Why is the function not considered a straight line according to the script?

The function is not a straight line because it has a discontinuity at x = 3, where the term x - 3 is canceled out, indicating that x cannot be equal to 3.

What is the y-intercept of the function as described in the script?

The y-intercept is 1/4, which is found by setting x to 0 in the simplified function.

What is the slope of the line when graphing the function from the script?

The slope of the line is 1/4, indicating that for every 1 unit increase in x, y increases by 1/4 unit.

How does the script suggest to handle the discontinuity at x = 3 when graphing the function?

The script suggests to mark x = 3 with an open circle on the graph and calculate the y-value at this point using the limit concept to understand the behavior of the function around 3.

What is the y-value at the point of discontinuity (x = 3) as per the script?

The y-value at x = 3 is 1, which is calculated by substituting x = 3 into the original function before simplification.

How is the range of the function described in the script?

The range of the function is all real numbers from negative infinity to 1, not including 1, and from 1 to infinity.

What is the significance of the open circle at (3, 1) on the graph as mentioned in the script?

The open circle at (3, 1) signifies that the point is not part of the function's graph because it corresponds to a discontinuity where the function is undefined.