Problem 9 - College Algebra

Question

Find the domain of each function. $$ f(x)-\frac{1}{x+7}+\frac{3}{x-9} $$

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \frac{1}{x+7}+\frac{3}{x-9}\) is all real numbers except -7 and 9.

Step 1: Define the domain

The domain of the function \(f(x)\) is all real numbers such that x does not make any denominator of the function equal to zero.

Step 2: Identify the undefined points

To find these points, set each denominator equal to zero and solve for x. For \(x+7=0\), solving for x we get \(x=-7\). For \(x-9=0\), solving for x we get \(x=9\).

Step 3: Define the domain of the function

The given function f(x) is undefined for x = -7 and x = 9. Hence, the domain of the function f(x) is all real numbers except -7 and 9.

Key Concepts

Understanding Rational Functions

Rational functions are a type of function that can be represented as the quotient of two polynomials. That means, if you have a function like \( f(x) = \frac{P(x)}{Q(x)} \), both \( P(x) \) and \( Q(x) \) are polynomials. The variable \( x \) is placed in both the numerator and the denominator.
This leads to a crucial property of rational functions: they can have values for \( x \) that make them undefined. This uniqueness comes from the fact that division by zero is not permitted, so any value that makes the denominator zero results in the function not having a real number value. This is what makes finding the domain of rational functions interesting.
To determine if a function is rational, verify if it can be expressed as one polynomial divided by another. If that's the case, it is indeed a rational function.

Identifying Undefined Points

Undefined points in a rational function occur where the denominator becomes zero because division by zero is impossible. To find these points, we set the denominator equal to zero and solve for \( x \). In the context of our example function \( f(x) = \frac{1}{x+7} + \frac{3}{x-9} \):

• The first term, \( \frac{1}{x+7} \), would become undefined when \( x+7=0 \), which gives us \( x=-7 \).
• The second term, \( \frac{3}{x-9} \), becomes undefined when \( x-9=0 \), resulting in \( x=9 \).

Thus, \( x \) values of -7 and 9 make the function undefined. These values are what we call the undefined points of the function.

All About Real Numbers in Domain

The term 'real numbers' refers to all possible numbers that can be represented on the number line, including both rational and irrational numbers. When determining the domain of a function, we often say it includes all real numbers except where the function is undefined.
In our case, the domain of \( f(x) = \frac{1}{x+7} + \frac{3}{x-9} \) includes all real numbers except -7 and 9 because those values make the denominators zero. To notate the domain properly, we use interval notation: it is \( (-\infty, -7) \cup (-7, 9) \cup (9, \infty) \).
Remember, for many functions like polynomials, the domain is all real numbers because they are always defined. However, for rational functions, some real numbers will not be included in the domain due to division by zero, as seen here.