91Ó°ÊÓ

Algebra plays a critical role in understanding piecewise functions. At its core, algebra deals with variables and the rules for manipulating these variables, like addition, subtraction, multiplication, and division.
Algebra can help us express relationships and solve equations that model real-world situations.
In the given exercise, we see a function split into two pieces.

• For each piece, algebraic expressions help define relationships between input (\(x\)) and output (\(y\)).
• The first piece, \(3-x\), applies when \(x < 0\). This expression utilizes subtraction, a primary algebraic operation.
• The second piece, \(3+2x-x^2\), is a more complex quadratic expression, employing addition, multiplication, and exponentiation (squaring).

These expressions allow us to evaluate the function by substituting different values of \(x\) according to the conditions set in the problem.

Inequalities guide us in determining which piece of a piecewise function to use. An inequality is a statement that compares two expressions and shows the relative size of the expressions. In the context of piecewise functions, inequalities establish the conditions that decide which algebraic expression to employ.
In this problem, the inequalities are:

• \(x < 0\)
• \(x \geq 0\)

These inequalities, respectively, dictate which part of the function applies.
- If \(x\) is negative (\(x < 0\)), you use the linear part, \(y=3-x\).- For non-negative \(x\) (\(x \geq 0\)), you switch to the quadratic part, \(y=3+2x-x^2\).
Inequalities like these help determine not only the operational scope of each part of the function but also the behavior and possible outputs depending on the critical values of \(x\).

Function evaluation involves substituting values into a function to determine the resulting outputs or \(y\)-values. For piecewise functions, this is a two-step process where you first check which condition of the piecewise function applies before performing the calculation.
In our exercise, you evaluate the function by deciding:

• For a specific \(x\), whether \(x < 0\) or \(x \geq 0\) holds. This directs you to the corresponding expression.
• Once the correct equation is identified, substitute \(x\) into this equation to find \(y\), the function's output.

For example:

• If \(x = -2\), since \(-2 < 0\), use \(y=3-(-2)=3+2=5\).
• If \(x = 1\), since \(1 \geq 0\), substitute into \(3 + 2(1) - 1^2 = 3 + 2 - 1 = 4\).

Function evaluation in piecewise contexts reinforces not just calculation but also understanding conditions and selecting the right expression efficiently.