Problem 42 - College Algebra

Question

Evaluate each piecewise function at the given values of the independent variable. $$h(x)=\left\\{\begin{array}{ccc}\frac{x^{2}-25}{x-5} & \text { if } & x \neq 5 \\\10 & \text { if } & x-5\end{array}\right.$$ $$a. h(7)$$ $$b. h(0)$$ $$c. h(5)$$

Step-by-Step Solution

Answer

The evaluated values are \(h(7) = 12\), \(h(0) = 5\), and \(h(5) = 10\).

Step 1 - Evaluating \(h(7)\)

First, check which rule applies when \(x = 7\). Since \(7 \neq 5\), the first rule applies here. Plugging the \(x\) value into the first rule of the function, we get \(h(7) = \frac{(7)^2 - 25}{7 - 5} = \frac{49 - 25}{2} = 12\). Note that the denominator does not become zero, which makes it a legal evaluation.

Step 2 - Evaluating \(h(0)\)

Again, check which rule applies for \(x=0\). As \(0\neq 5\), the first rule applies. Substituting \(x = 0\) into the first rule, we get \(h(0) = \frac{(0)^2 - 25}{0 - 5} = \frac{-25}{-5} = 5\). The denominator does not become zero so it's a valid evaluation.

Step 3 - Evaluating \(h(5)\)

Here, the value of \(x\) is \(5\), therefore the second rule of the function applies. From the function definition, directly we get \(h(5) = 10\).

Key Concepts

Evaluating Functions

Evaluating functions means calculating the output of a function based on a given input. In the context of piecewise functions, different rules or expressions are applied depending on the value of the input variable (commonly represented by \(x\)). To evaluate a piecewise function at particular values, you follow these steps:

• Identify which rule of the piecewise function applies to the given input \(x\).
• Substitute the input value into the correct rule or expression.
• Simplify the expression to find the output of the function.

Remember, it's crucial to check the conditions of each rule first. These conditions act like gates, directing you to the correct piecewise rule that matches your input. For example, with the function \(h(x)\), when \(x = 7\), the condition \(x eq 5\) is satisfied, so you use the first rule. Evaluating these functions step-by-step ensures your calculations adhere to their proper conditions.

Independent Variables

The independent variable in a function is the input value that you control or choose, often represented as \(x\). It is the variable whose value is applied to the function, determining the function's output based on the given expression or rule.In piecewise functions, the independent variable is crucial as it dictates which specific function rule to apply. For the function \(h(x)\), \(x\) serves as the independent variable. Depending on whether \(x eq 5\) or \(x = 5\), you apply different parts of the piecewise function.Understanding the role of the independent variable helps in navigating through a piecewise function's multiple expressions. It allows you to confidently substitute the chosen value into the appropriate part of the function and accurately solve for the output.

Function Rules

Function rules define the operations or calculations performed in a function to produce an output from an input. In piecewise functions, there are typically multiple rules, each governing different ranges or specific values of the input.

• Each section of a piecewise function can be seen as a separate mini-function, operating based on its own "rule."
• For instance, in \(h(x)\), the rule \(\frac{x^{2} - 25}{x - 5}\) applies only when \(x eq 5\). Meanwhile, the rule \(10\) applies exactly when \(x = 5\).

It's essential to pay close attention to the conditions attached to each rule. These conditions clarify when each rule should be applied, ensuring that the function is evaluated correctly for any given input. Understanding function rules in piecewise functions allows you to find the correct segment and calculate accurately, without confusion.