How to Find the Domain and Range of Reciprocal Functions

TL;DR

To find the domain of a reciprocal function, exclude the vertical asymptote from all x-values, resulting in an interval notation of (-∞, h) ∪ (h, ∞). For the range, exclude the horizontal asymptote from all y-values, giving a range of (-∞, k) ∪ (k, ∞). Understanding these exclusions is crucial for accurately determining the function's domain and range.

Transcript

in this video we're going to talk about how to find the domain and range of a reciprocal function so here is the standard form of a reciprocal function y is equal to a / x - h plus K now what I'm going to do is I'm going to draw a rough sketch of this function and to do that I need to determine the horizontal ASM toote and the vertical ASM toote th... Read More

Key Insights

• ❣️ The standard form of a reciprocal function is y = a / (x - h) + k.
• 🔃 The horizontal asymptote is determined by the value of k, while the vertical asymptote is determined by the value of h.
• ☺️ The domain of a reciprocal function includes all x values except for the vertical asymptote.
• 😀 The range of a reciprocal function includes all y values except for the horizontal asymptote.
• ❎ Reciprocal functions with positive values of a exist in quadrants 1 and 3, while those with negative values of a exist in quadrants 2 and 4.
• 🧡 It is important to remove the vertical asymptote from the domain and the horizontal asymptote from the range when writing them in interval notation.

Questions & Answers

Q: How do you find the horizontal asymptote of a reciprocal function?

The horizontal asymptote of a reciprocal function is determined by the value of k in the equation y = a / (x - h) + k. If k is positive, the horizontal asymptote is at y = k, and if k is negative, the asymptote is at y = k.

Q: What is the significance of the vertical asymptote in a reciprocal function?

The vertical asymptote in a reciprocal function is determined by the value of h in the equation y = a / (x - h) + k. It represents the value of x for which the function is undefined, as it results in division by zero.

Q: How do you write the domain of a reciprocal function?

To write the domain, you need to include all x values except for the vertical asymptote. In interval notation, the domain would be (-∞, 2) ∪ (2, ∞).

Q: How do you write the range of a reciprocal function?

To write the range, you need to consider the y values of the function. The range includes all y values except for the horizontal asymptote. In interval notation, the range would be (-∞, 4) ∪ (4, ∞).

Summary & Key Takeaways

• Reciprocal functions can be represented by the equation y = a / (x - h) + k.

• The horizontal asymptote is determined by the value of k, while the vertical asymptote is determined by the value of h.

• The domain of a reciprocal function includes all x values except for the vertical asymptote, while the range includes all y values except for the horizontal asymptote.