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Chapter 2: Problem 82

Find the domain of the function given by each equation. $$ f(x)=\frac{7}{6-x} $$

Short Answer

Expert verified
The domain is \((-\text{鈭瀩, 6) \cup (6, \text{鈭瀩)\).

Step by step solution

01

- Identify the Denominator

The function given is \[f(x)=\frac{7}{6-x}\]. Identify the denominator of the fraction, which is \(6-x\).
02

- Set the Denominator Not Equal to Zero

For the function to be defined, the denominator cannot be zero. So, set \(6-x eq 0\).
03

- Solve the Inequality

Solve for \(x\): \[6-x eq 0\] \(x eq 6\).
04

- State the Domain

The domain of the function is all real numbers except \(6\). In interval notation, this is written as \((-\text{鈭瀩, 6) \cup (6, \text{鈭瀩)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Functions
A rational function is a type of function that can be expressed as the ratio of two polynomials. It typically looks like this: \ \ \(f(x) = \frac{P(x)}{Q(x)}\) \In this example, our function \ \ \(f(x)=\frac{7}{6-x}\) is a rational function because it is the ratio of the constant 7 over the linear polynomial \ \(6-x\). \Rational functions can have many interesting features: \
  • Asymptotes: Lines that the graph of the function approaches but never touches.
  • Holes: Points where the function is undefined due to the denominator being zero.
  • Intercepts: These are the points where the function crosses the x-axis or y-axis.
Understanding rational functions is essential for working through domains, ranges, and various properties of the functions.
Interval Notation
Interval notation is a mathematical method to describe a set of numbers along a number line. It's particularly useful for expressing the domain and range of functions. \Rather than writing out limits in long form, interval notation uses brackets and parentheses to succinctly specify bounds: \
  • (a, b) denotes all numbers between a and b, not including a and b.
  • [a, b] denotes all numbers between a and b, including both a and b.
  • (a, b] denotes all numbers between a and b, including b but not a.
  • [a, b) denotes all numbers between a and b, including a but not b.
  • (-\text{鈭瀩, a) means all numbers less than a, while (a, \text{鈭瀩) means all numbers greater than a.
In the context of our function \ \(f(x) = \frac{7}{6-x}\), the domain excludes the value x=6 since it makes the denominator zero. Thus, using interval notation, the domain is expressed as \ \((-\text{鈭瀩, 6) \cup (6, \text{鈭瀩)\).
Denominator
The denominator is the bottom part of a fraction that tells us into how many equal parts the whole is divided. It's crucial when dealing with rational functions since the denominator can impact where the function is defined. \For our function \ \(f(x) = \frac{7}{6-x}\), identification and manipulation of the denominator \ \(6-x\) are essential steps to determine the domain. Here's how: \
  • Identify the Denominator: In our function, the denominator is \ \(6-x\).
  • Find the Restrictions: The function is undefined when the denominator is zero. So we set up the equation \ \(6-xeq0\).
  • Solve for Restrictions: Solve the equation to find that \ \(xeq6\). This tells us that at x=6, the function is undefined.
The step-by-step approach to managing the denominator helps in accurately determining where the function exists and how to express these regions using interval notation.

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