Problem 25 - Algebra Form and Function

Question

Use the table to fill in the missing values. (There may be more than one answer.) (a) \(h(0)=?\) (b) \(h(?)=0\) (c) \(h(-2)=?\) (d) \(h(?)=-2\) $$ \begin{array}{c|c|c|c|c|c|c|c} \hline t & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline h(t) & -1 & 0 & -3 & -2 & -1 & -2 & 0 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Answer: h(0) = -2

Step 1: (a) Find h(0)

To find h(0), look for the cell that corresponds to the input value t=0 in the table. We can see that when t=0, h(t) is given to be -2. So, h(0) = -2.

Step 2: (b) Find the input value when h(t) = 0

To find the input value(s) when h(t)=0, look for the cell(s) that corresponds to the output h(t)=0 in the table. We can see that there are two instances - when t=-2 and when t=3. So, the input values are -2 and 3.

Step 3: (c) Find h(-2)

To find h(-2), look for the cell that corresponds to the input value t=-2 in the table. We can see that when t=-2, h(t) is given to be 0. So, h(-2) = 0.

Step 4: (d) Find the input value when h(t) = -2

To find the input value(s) when h(t)=-2, look for the cell(s) that corresponds to the output h(t)=-2 in the table. We can see that there are two instances - when t=0 and when t=2. So, the input values are 0 and 2.

Key Concepts

Function Evaluation

Evaluating a function is about finding the output for a given input using a predefined rule. In algebra, functions are often defined by their expressions that relate their inputs (often denoted as \(t\)) to outputs (denoted as \(h(t)\)). Each input should map to one specific output according to the function's rule.

To evaluate a function with a particular input, you simply plug that input into the function. For example, if the problem asks you to find \(h(0)\), you need to see what output the function assigns to the input 0. In this instance, using the table provided, we found that \(h(0) = -2\). The table serves as a representation of the function where inputs are mapped to outputs, making evaluation straightforward.

• Identify the input from the table.
• Locate the corresponding output value.
• Confirm the association with the given rule or expression if any.

Input-Output Table

Input-output tables are a helpful way to visually represent a function. They show how each input value connects to its respective output, eliminating complex calculations and making function evaluation more intuitive.

The table provided in this problem lists values of \(t\) and their corresponding outputs \(h(t)\) row-wise:- When \(t = -3\), \(h(t) = -1\)- When \(t = -2\), \(h(t) = 0\)- When \(t = 0\), \(h(t) = -2\), and so on.

This structured approach helps to quickly locate information:

• Identify the input on one row of the table.
• Look directly across to find its output.
• This method also allows for reverse tracing when you have an output and need the original input.

The ability to scan, locate and interpret such tables is a key skill in solving algebraic functions and ensures clarity in function evaluations.

Step-by-Step Solutions

Using a step-by-step approach is crucial for understanding how to solve function-related problems thoroughly. It breaks down complex tasks into manageable steps, ensuring each part of the solution is clear and comprehensive.

Start by understanding the specific question, like finding \(h(0)\) or identifying for what \(t\) is \(h(t) = 0\). Each step should:

• Start with identifying the task—either finding an input or output.
• Examine the table to find the necessary intersecting value.
• Confirm the correctness at each step to avoid mistakes.

For instance, to find \(h(-2)\), you would follow these steps:- Recognize the input value, which is \(-2\).- Locate \(-2\) in the table's input row.- Trace across that row to find the output, \(0\).

By approaching problems one step at a time, you not only enhance accuracy but also build a deeper understanding of the function, making future evaluations easier and less prone to errors.