91Ó°ÊÓ

When dealing with piecewise functions, function evaluation involves determining which formula to use. In our example, the function is defined as:

• If \(x < 3\), use the piece \(f(x) = x - 2\)
• If \(x \geq 3\), use the piece \(f(x) = 5 - x\)

To evaluate a piecewise function, first, check which piece of the function applies. The value of \(x\) guides your choice. Then, simply substitute the given \(x\) value into the relevant piece. For instance, to evaluate \(f(-5)\), we see that \(-5 < 3\), so we use \(f(x) = x - 2\) and substitute to get \(f(-5) = -5 - 2 = -7\). This process allows you to calculate function values systematically, even without a calculator. Each piece of the function applies under specific conditions—mastering this makes function evaluation straightforward.

Inequalities are mathematical statements that show the relationship between two values. They express whether one value is less than, greater than, or equal to another. In piecewise functions, inequalities define which piece of the function to use for a given \(x\). For example,

• The inequality \(x < 3\) guides us to use \(f(x) = x - 2\).
• The inequality \(x \geq 3\) directs us to use \(f(x) = 5 - x\).

By understanding the inequalities, you can quickly decide where an \(x\) value falls and which formula to apply. This direct matching of \(x\) with conditions helps avoid mistakes in selecting the wrong part of the function, ensuring correct evaluations. Learning to read and interpret inequalities is crucial as they control function evaluations and are a common element in algebraic expressions.

Algebraic expressions are combinations of numbers, variables, and mathematical operations. In our piecewise function example, the expressions used are:

• \(f(x) = x - 2\) for \(x < 3\)
• \(f(x) = 5 - x\) for \(x \geq 3\)

These expressions involve simple arithmetic operations: subtraction of a constant from \(x\) or subtraction of \(x\) from a constant. To solve these, substitute the values directly into the expression, replacing \(x\) with the numbers provided. For instance, substituting \(3\) into \(f(x) = 5 - x\) gives \(f(3) = 5 - 3 = 2\). Understanding these fundamental operations within algebraic expressions is essential for solving various mathematical problems and effectively using piecewise functions.