Question

Find the domain of the function. $$ f(x)=\frac{1}{|x+3|} $$

Short answer

The domain of the function is all real numbers except -3.

Step-by-step solution

Identify the denominator

The denominator in the function is \( |x+3| \). This is the absolute value of \( x+3 \).

Solve for x

The denominator becomes zero when \( x+3=0 \). Solving for x we get \( x=-3 \).

Formulate the Domain

From the solution, we see that x=-3 makes the denominator zero and thus undefined. The domain of the function will therefore be all real numbers except -3.

Key concepts

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, irrespective of the direction. It is always a non-negative value. In mathematical notation, the absolute value of a number 'a' is denoted by |a|. For example, both |3| and |-3| equal 3 because each is three units away from zero on the number line.

When dealing with functions, the absolute value can affect the domain because it impacts the output values of the function. Functions with absolute values in the denominator, like in the exercise with the function \( f(x) = \frac{1}{|x+3|} \), require special attention because the absolute value expression cannot equal zero, as division by zero is undefined.

Denominator of a Function

In a fraction, the denominator represents the number of equal parts the whole is divided into, and it is found below the fraction bar. When examining a function that involves a fraction, the denominator indicates the divisor function. For example, in the function \( f(x) = \frac{1}{|x+3|} \), '|x+3|' is the denominator.

The crucial rule to remember is that the denominator of a function cannot be zero because division by zero is not defined in mathematics. When determining the domain of a function, it is essential to identify values that would make the denominator zero and exclude them from the domain, ensuring that all function values remain defined.

Undefined Function Values

Function values become undefined when an operation within the function does not produce a real number. One common scenario where this occurs is division by zero. Since any number divided by zero does not have a well-defined value, a function that would require such division is considered undefined at that point.

For instance, in our textbook exercise, the function \( f(x) = \frac{1}{|x+3|} \) will be undefined when the absolute value expression in the denominator, '|x+3|', is equal to zero, which happens when x equals -3. To define the domain properly, we must exclude x = -3 from the set of possible x-values, because at x = -3, the function does not produce a real number. The domain is thus all real numbers except x = -3.