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Chapter 3: Problem 82

Find the domain of \(f\). $$f(x)=\frac{7}{6-x}$$

Short Answer

Expert verified
The domain of the function is (-鈭�, 6) 鈭� (6, 鈭�).

Step by step solution

01

Identify the function type

Recognize that the given function is a rational function: note that the numerator is 7 and the denominator is 6-x.
02

Find restrictions on the denominator

A rational function is undefined when its denominator is zero. Set the denominator equal to zero and solve for x: 6 - x = 0.
03

Solve the restriction

Solve for x in the equation found in Step 2: 鈥�6 - x = 0 鈥� x = 6.
04

Determine the domain

Since the function is undefined at x = 6, exclude 6 from the domain. The domain of the function is all real numbers except for this value, represented as: 鈥�(-鈭�, 6) 鈭� (6, 鈭�).

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Key Concepts

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

Rational Functions
Rational functions are a type of function where one polynomial is divided by another polynomial. In general, a rational function can be written as:
\[ f(x) = \frac{P(x)}{Q(x)} \]
Here, \( P(x) \) and \( Q(x) \) are polynomials, and the function is undefined wherever \( Q(x) = 0 \).
In the given exercise, the rational function is:
\[ f(x) = \frac{7}{6-x} \]
This is a simple rational function where the numerator is 7 (a constant polynomial), and the denominator is \( 6-x \).
Domain Restrictions
The domain of a function is the set of all possible input values (x-values) that the function can accept. For rational functions, the primary restriction comes from the denominator. The denominator cannot be zero because division by zero is undefined.
In the function \[ f(x) = \frac{7}{6-x}, \] we need to exclude values of \( x \) that make the denominator zero. To find these values, set the denominator equal to zero:
\[ 6 - x = 0 \]
Solving for \( x \) gives: \[ x = 6 \]
So, the domain of \( f(x) \) is all real numbers except \( x = 6 \). In interval notation, this is represented as:
\[ (-鈭�, 6) 鈭� (6, 鈭�) \]
Solving Equations
Solving equations is a fundamental skill in mathematics. It involves finding the value(s) of the variable that satisfy the equation. In our case, we solved the equation to find the domain restriction. Let's revisit the steps:
  • Set the denominator equal to zero: \( 6 - x = 0 \)
  • Solve for \( x \): \( x = 6 \)
This tells us that \( x = 6 \) makes the denominator zero, meaning \( f(x) \) is undefined at this point.
Properly solving the equation helped us identify the domain restriction, ensuring we understand which values are not allowed in the function's domain.

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