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An algebraic function is a function that can be defined using basic algebraic operations such as addition, subtraction, multiplication, and division, as well as taking roots. These functions are typically composed of polynomials and can include variables with exponents.
In our exercise, the function given is: \[ f(x) = x + |x| \]
This is an algebraic function because it involves basic algebraic operations and the absolute value of a variable. Understanding how to manipulate these types of functions is essential for solving various kinds of problems in math.

Key points to consider:

• Algebraic operations are the building blocks of algebraic functions.
• They can be visualized using graphs.
• Knowing how algebraic functions behave helps solve equations and inequalities.

The absolute value function, denoted as \(|x|\), measures the distance of a number from zero on the number line, ignoring direction. Therefore, \(|x|\) is always non-negative. For any real number \(x\), \(|x| = x\) if \(x \geq 0\), and \(|x| = -x\) if \(x < 0\).
In our function, \(f(x) = x + |x|\), understanding the absolute value is crucial because it affects how we interpret \(f(-x)\).
When substituting \(-x\) into the absolute value function:
\( f(-x) = -x + |-x| = -x + |x| \)
This reveals that the function combines linear and absolute value terms, and their interaction depending on the sign of \(x\) will determine the overall behavior of \(f(x)\).

The symmetry of a function helps in determining whether a function is even, odd, or neither. For a function \(f(x)\):
**Even Function:** \(f(x) = f(-x)\). This implies that the function is symmetric about the y-axis.
**Odd Function:** \(f(-x) = -f(x)\). This implies that the function has rotational symmetry about the origin.
**Neither:** If a function does not satisfy either condition, it is neither even nor odd.
For our function:
\[ f(x) = x + |x| \text{ and } f(-x) = -x + |x| \]
Clearly, \(f(x)\) does not equal \(f(-x)\) and \(f(-x)\) does not equal \(-f(x)\), indicating the function is neither even nor odd. Understanding such symmetry can be valuable for graphing and integration.
Key points:

• Even functions have y-axis symmetry.
• Odd functions have origin symmetry.
• Symmetry is a powerful tool in solving and simplifying problems.

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values). For our function \(f(x) = x + |x|\), both domain and range are straightforward.

The **domain** of \(f(x)\) is all real numbers (\(\textbf{R}\)) because we can substitute any real number for \(x\) in the expression \(x + |x|\).
The **range** of \(f(x)\) also turns out to be all real numbers (\(\textbf{R}\)), because the combination of linear and absolute value terms can produce any real number output depending on positive or negative inputs.

• The domain gives us the permissible x-values.
• The range tells us the possible y-values.
• Understanding these concepts ensures a function is well-defined and helps in graphing it correctly.