Problem 54 - College Algebra

Question

For the following exercises, write the domain for the piecewise function in interval notation. $$ f(x)=\left\\{\begin{array}{ll}{2 x-3} & {\text { if } x<0} \\ {-3 x^{2}} & {\text { if } x \geq 2}\end{array}\right. $$

Step-by-Step Solution

Answer

The domain of the function is \((-\infty, 0) \cup [2, \infty)\).

Step 1: Understand the Definition of the Piecewise Function

The given function is a piecewise function, which means it is defined by different expressions depending on the value of \(x\). There are two expressions: one for \(x < 0\) and another for \(x \geq 2\).

Step 2: Determine the Domain of Each Piece

For the first piece, \(f(x) = 2x - 3\) when \(x < 0\), the domain is all \(x\) less than 0, which in interval notation is \((-\infty, 0)\). For the second piece, \(f(x) = -3x^2\) when \(x \geq 2\), the domain is all \(x\) greater than or equal to 2, which in interval notation is \([2, \infty)\).

Step 3: Combine the Domains

Since there are no overlapping intervals and each piece covers different parts of the real number line, the overall domain of the function is the union of the domains of the individual pieces: \((-\infty, 0) \cup [2, \infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial in mathematics. It tells us all the possible input values for which the function is defined. In simpler words, it's the set of all x-values that can go into a function without causing any issues, such as division by zero or taking the square root of a negative number (in the context of real numbers).

For piecewise functions, like the one in the exercise, each 'piece' of the function has its domain. Here we have two pieces:

• The first piece, \(2x - 3\), is defined when \(x < 0\). So for this piece, the domain is all x-values less than zero.
• The second piece, \(-3x^2\), is defined when \(x \geq 2\). Thus, its domain includes all x-values that are two or more.

To find the total domain of this piecewise function, we simply combine the domains from all pieces where they are applicable. The piecewise nature means that there might not be one continuous interval; there can be gaps. Here, these intervals just need to be joined without an overlap, resulting in domains like union sets.

Interval Notation

Interval notation is a mathematical shorthand to express the set of all numbers between given lower and upper limits. It clearly depicts which numbers are included in the range and which are not, using parentheses \(()\) and square brackets \([]\).

- Parentheses, as in \((a, b)\), mean that a and b are not included in the interval.- Square brackets, as in \([a, b]\), mean that a and b are included.

For specific scenarios like our exercise:

• The interval \((-\infty, 0)\) means all numbers less than 0, not including 0 itself.
• The interval \([2, \infty)\) means all numbers from 2, including 2, and extending indefinitely to greater numbers.

Interval notation is especially useful when communicating the domain of a function as it condenses complex information into a simple and clear format.

Real Number Line

The real number line is an essential concept in understanding domains in mathematics. It represents all possible real numbers as points on one infinite line. Each point on this line correlates to a real number, which includes both rational and irrational numbers.

Understanding the real number line helps when determining domains of functions because it directly shows where a function is applicable or not. This is crucial for piecewise functions, like in our exercise, where different parts of the function are defined over different segments of the number line.

When we talk about intervals, we are selecting a section of the real number line. For example:

• An interval like \((-\infty, 0)\) slices off a portion to include all numbers less than 0.
• Similarly, \([2, \infty)\) includes numbers starting at 2 and extending infinitely.

Visualizing these intervals as segments or points on the number line can make it easier to grasp which set of inputs a function can accept.