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Chapter 3: Problem 113

Explain why the domain of the function given by \(f(x)=\frac{x+3}{2}\) is \(\mathbb{R},\) but the domain of the function given by \(g(x)=\frac{2}{x+3}\) is not \(\mathbb{R}\).

Short Answer

Expert verified
The domain of \(f(x)\) is \(\mathbb{R}\) because there are no restrictions. The domain of \(g(x)\) is \(\mathbb{R} \setminus \{-3\}\) because \(x=-3\) causes division by zero.

Step by step solution

01

- Define the Domain of a Function

The domain of a function is the set of all possible input values (x-values) that will produce a valid output (y-value) for the function. To determine the domain, identify any restrictions on the input values.
02

- Analyze the Function f(x)

For the function \(f(x)=\frac{x+3}{2}\), you can observe that it is a linear function. Since there are no restrictions (like division by zero or taking the square root of a negative number), the domain of \(f(x)\) is all real numbers (\(\mathbb{R}\)).
03

- Analyze the Function g(x)

For the function \(g(x)=\frac{2}{x+3}\), there is a restriction because the denominator cannot be zero. Set the denominator equal to zero to find the restriction: \(x+3=0\).
04

- Determine the Restricted Value for g(x)

Solving the equation \(x+3=0\), we get \(x=-3\). Therefore, \(x=-3\) is the value that would make the denominator zero, which is not allowed.
05

- State the Domain of g(x)

Since \(x=-3\) is not allowed, the domain of \(g(x)\) includes all real numbers except \(x=-3\). It can be written as \(\mathbb{R} \setminus \{-3\}\).

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Key Concepts

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

Input Values
When we talk about the domain of a function, it primarily refers to the input values. These are the values of the independent variable (often represented as 'x') that you can plug into the function. The set of input values includes every number you can use to get a meaningful output from the function.

To better understand, consider the function examples provided:
  • For the function \(f(x) = \frac{x + 3}{2}\), you can insert any real number as 'x'. No matter which number you choose, you will always get a valid output because there's no special condition restricting your input values.
  • On the other hand, for the function \(g(x) = \frac{2}{x + 3}\), you need to be mindful about the value that makes the denominator zero.
Restrictions
The concept of restrictions in mathematical functions essentially means identifying the values that can cause the function to be undefined or problematic.

Restrictions often occur in situations such as:
  • Division by zero: This is the most common type of restriction. If the denominator of a fraction equals zero, the function becomes undefined.
  • Square root of a negative number: In the domain of real numbers, we avoid taking the square root of negative numbers, as they do not yield real results.
  • Logarithms of non-positive numbers: Similar to square roots, logarithms have their restrictions, as they are not defined for zero or negative inputs.

For our case, the key restriction to note comes from the function \[ g(x) = \frac{2}{x+3} \]. The denominator should not be zero, because division by zero is not allowed.
Division by Zero
One of the most critical restrictions involves division by zero. In mathematics, division by zero is undefined because it does not produce a finite or meaningful result.

Let's examine the problematic areas for our function examples:
  • For \[ f(x) = \frac{x + 3}{2} \], the denominator is a constant value (2) and will never be zero. Therefore, no restriction due to division by zero applies. The domain of this function is all real numbers: \[ \text{Domain of } f(x) = \text{R} \].
  • For \[ g(x) = \frac{2}{x + 3} \], we must consider when the denominator equals zero, i.e., \[ x + 3 = 0 \]. Solving this gives \[ x = -3 \]. So, we restrict \[ x = -3 \]. The domain of this function includes all real numbers except \[ x = -3 \]: \[ \text{Domain of } g(x) = \text{R} \backslash \text{-3} \].

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