Problem 49 - College Algebra

Question

Evaluate each piecewise function at the given values of the independent variable. $$h(x)=\left\\{\begin{array}{cl}\frac{x^{2}-9}{x-3} & \text { if } x \neq 3 \\\ 6 & \text { if } x=3\end{array}\right.$$ a. \(h(5)\) b. \(h(0)\) c. \(h(3)\)

Step-by-Step Solution

Answer

The solutions are: a. \(h(5) = 8\), b. \(h(0) = 3\), c. \(h(3) = 6\).

Step 1: Evaluate \(h(x)\) at \(x = 5\)

According to the definition of the function, when \(x\neq 3\), \(h(x)\) is defined as \(\frac{x^{2}-9}{x-3}\). Now, substituting \(x = 5\) into this part of the function, we get \(h(5) = \frac{5^{2}-9}{5-3} = \frac{16}{2} = 8.\)

Step 2: Evaluate \(h(x)\) at \(x = 0\)

Following similar steps, we replace \(x\) with \(0\) into the first part of the function definition. So, \(h(0) = \frac{0^{2}-9}{0-3} = \frac{-9}{-3} = 3.\)

Step 3: Evaluate \(h(x)\) at \(x = 3\)

For \(x = 3\), the function is directly defined as \(6\). Thus, \(h(3) = 6.\)

Key Concepts

Evaluating Functions

Understanding how to evaluate piecewise functions is like being a detective; you must examine each case separately to reveal the function's true value. A piecewise function is simply an algebraic expression broken into pieces, with each piece having its own specific rule for calculation. The value is determined based on the input, which is where evaluation comes into play.

Consider evaluating the piecewise function at a particular value of the independent variable, such as a number like 5 or a specific condition like 'if x equals 3.' To successfully evaluate it, we apply the rules that correspond to the given value of the independent variable. It's akin to following a map that splits into different paths; each path leads to a different outcome.

In the case of our example function, for any value except 3, you'll use the first rule, which is an algebraic fraction. However, when the value is precisely 3, you automatically use the second rule, which assigns a constant value to the function. Evaluating functions correctly is fundamental to understanding their behavior at different points and is a crucial aspect of learning how to graph them.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations. Think of algebraic expressions as a secret code; combining numbers and variables together in certain ways tells us something unique each time. When we’re working with functions, particularly piecewise ones, these expressions define how the function behaves over different intervals.

For instance, take the expression \(\frac{x^2-9}{x-3}\). It's an algebraic expression involving subtraction, exponentiation, and division. These thrive at the heart of algebra and allow us to explore relationships between variables.

Why Simplify?

It's often useful to simplify an expression by combining like terms or factoring. This makes it easier to evaluate or can reveal more information about the function, such as potential discontinuities. Simplification isn't just cosmetic; it helps us peer deeper into the mathematics and aids in understanding the underlying structure of our equation.

Function Definitions

Defining functions, especially piecewise functions, is akin to drafting a rulebook for how a function plays the game of mathematics on the coordinate grid. It's how we tell anyone, or any calculator, what operations to perform based on the input value. In algebra, these rules are crucial for understanding and managing functions.

A piecewise function is defined by two or more expressions, each applying to a particular part of the function's domain, or the set of all possible input values. It creates a mathematical patchwork, and knowing each piece's applicability is key to correctly applying the function as a whole. Function definitions are the guideposts that help align our thoughts and calculations to correctly evaluate functions at any given point.

Effective communication of these definitions is fundamental to algebra. A clearly defined function prevents misunderstandings and provides a clear protocol for evaluation, ensuring that students and mathematicians can consistently predict the behavior of the function across its entire domain.