Problem 57 - College Algebra

Question

Find the domain of each function. $$f(x)=\frac{2}{(x+3)(x-7)}$$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{2}{(x+3)(x-7)}\) is all real numbers except -3 and 7. It can be described in interval notation as \( (-\infty, -3) \cup (-3, 7) \cup (7, +\infty) \).

Step 1: Identify the Denominator

The function in question is a rational function, and it's written in the form \(f(x) = \frac{A}{B}\), where both A and B are polynomials. Looking at our function, we seek to find the domain, and we notice that the denominator (B) is equal to \((x+3)(x-7)\). Remember, the function is undefined where the denominator equals zero.

Step 2: Solve for x-values that make the denominator zero

To find the values that will make the denominator zero, we solve for x in the denominator equation. That is, we solve the equations \(x+3=0\) and \(x-7=0\). These are obtained from each factor in our denominator. Solving for x, we get x = -3 and x = 7.

Step 3: Define the domain of the function

The domain of this function will be all real numbers except -3 and 7 because these values make the denominator equal to zero and thus make the function undefined. Therefore, our domain can be described in interval notation as \( (-\infty, -3) \cup (-3, 7) \cup (7, +\infty) \).

Key Concepts

Rational Functions

Rational functions are a type of function expressed as the ratio of two polynomials. It can be represented in the general form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials.
The most important thing to note about rational functions is that they are only defined when the denominator \( Q(x) \) is not equal to zero. This is because division by zero is undefined in mathematics.
In the case of our function \( f(x) = \frac{2}{(x+3)(x-7)} \), the numerator is 2, and the denominator is \( (x+3)(x-7) \). The domain of the function will include all real numbers except those which make the denominator equal to zero.
When dealing with rational functions:

• Check the denominator to identify values that make it zero.
• Exclude these values from the domain.

Undefined Expressions

An expression is considered undefined for certain values when it involves division by zero. In mathematics, division by zero doesn't have a meaning and is therefore considered undefined.
In the context of the given rational function \( f(x) = \frac{2}{(x+3)(x-7)} \), we must determine which values of \( x \) will make \((x+3)(x-7) = 0\). This implies finding where each factor inside the denominator equals zero.
By setting \( x+3 = 0 \) and \( x-7 = 0 \), we can solve to find \( x = -3 \) and \( x = 7 \). These specific x-values make the expression undefined.

• Identify expressions in the denominator that cause division by zero.
• Solve each expression for zero to find the problem points.

Interval Notation

Interval notation is a concise way to describe subsets of the real number line. It is especially useful for defining the domain of functions such as rational functions, where certain x-values are excluded.
In interval notation:

• Parentheses \( () \) indicate that the endpoint is not included in the interval.
• Brackets \( [] \) would mean that an endpoint is included, but they are not used here because our endpoints correspond to undefined values (\( x = -3 \) and \( x = 7 \)).

Using interval notation, the domain of the function \( f(x) = \frac{2}{(x+3)(x-7)} \) is written as \((-\infty, -3) \cup (-3, 7) \cup (7, \infty)\). This notation shows the parts of the number line where the function is defined, concatenated with the union \( \cup \) symbol, which represents a collection of separate intervals.