Problem 26 - Algebra Form and Function

Question

Use the table to fill in the missing values. (There may be more than one answer.) (a) \(h(?)=2 h(0)\) (b) \(h(?)=2 h(-3)+h(2)\) (c) \(h(?)=h(-2)\) (d) \(h(?)=h(1)+h(2)\) $$ \begin{array}{c|c|c|c|c|c|c|c} t & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline h(t) & -1 & 0 & -4 & -2 & -1 & -2 & 0 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

)=2 h(0) b) h(?)=2 h(-3)+h(2) c) h(?)=h(-2) d) h(?)=h(1)+h(2) Answer: a) h(-1)=2 h(0) b) h(-1)=2 h(-3)+h(2) c) h(-2)=h(-2) d) No solution.

Step 1: Identify h(0) in the table

Looking at the table, h(0) = -2.

Step 2: Calculate 2 h(0)

Multiply h(0) by 2: 2 * (-2) = -4 Now we need to look for the value -4 in the table to identify what value of t corresponds to it. -4 corresponds to t = -1 in the table. #a) Result#: \(h(-1) = 2 h(0)\) #b) h(?)=2 h(-3)+h(2)#

Step 3: Identify h(-3) and h(2) in the table

Looking at the table, h(-3) = -1 and h(2) = -2.

Step 4: Calculate 2 h(-3) + h(2)

Substitute the known values in the equation: 2 * (-1) + (-2) = -4 Now, we look for the value -4 in the table. -4 corresponds to t = -1 in the table. #b) Result#: \(h(-1) = 2 h(-3) + h(2)\) #c) h(?)=h(-2)#

Step 5: Identify h(-2) in the table

Looking at the table, h(-2) = 0. Since h(-2) = 0, the missing value should correspond to t = -2. #c) Result#: \(h(-2) = h(-2)\) #d) h(?)=h(1)+h(2)#

Step 6: Identify h(1) and h(2) in the table

Looking at the table, h(1) = -1 and h(2) = -2.

Step 7: Calculate h(1) + h(2)

Add h(1) and h(2): -1 + (-2) = -3 Now, we look for the value -3 in the table. However, there is no value of h(t) equal to -3 in the table, so there is no solution for this expression. #d) Result#: No solution.

Key Concepts

function_evaluation

Function evaluation is the process of determining the output of a function for a particular input value. In mathematical terms, a function is a rule that assigns each input from a set of inputs (called the domain) to exactly one output in a set of possible outputs (called the range). To evaluate a function means to find the value of the function's output for a specific input. Let's consider the function given by the table in the exercise. For each value of \( t \), there is a corresponding \( h(t) \). For example:

• When \( t = -3 \), \( h(t) = -1 \).
• When \( t = 0 \), \( h(t) = -2 \).

To solve these problems, you're essentially performing function evaluation by locating the specific values in the table, as seen in the step-by-step solution. Evaluating involves not only finding individual function values but combining them mathematically—like multiplying or adding them—to solve for unknown variables.

function_table

A function table is a representation of outputs based on corresponding inputs. Each row or column in a function table, like the one provided in the exercise, usually represents a specific input-output pairing. This exercise gives a set of values for \( t \) and their corresponding outputs \( h(t) \):

• The table lists values for \(-3 \le t \le 3 \).
• The corresponding \( h(t) \) values vary based on \( t \).

Function tables allow you to find and verify function relationships with ease. For example, if you need to solve for \( h(?) = 2h(0) \), you simply find \( h(0) \) in the table, perform calculations, and then locate the resulting value in the same table to find the corresponding \( t \). This table-driven method is a practical way to evaluate and solve function-related questions.

problem_solving

Problem-solving in the context of functions often means understanding how to manipulate and use function formulas and values to find unknown quantities. In this exercise, you must use known values from a function table to deduce and solve for \( t \) values that satisfy different function equations. The process involves:

• Identifying known values from the table, such as \( h(0) = -2 \) and \( h(1) = -1 \).
• Applying these values in new expressions like \( h(?) = 2h(0) \) to re-calculate and solve for specific \( t \).
• Checking the computed results against the table to verify accuracy.

Problem-solving with functions can involve trial and error, logical deduction, and arithmetic applications to ensure you find the correct solutions.

function_identification

Function identification involves recognizing specific functions and their properties by using given information or structures, like tables or equations. It's about matching given conditions to known function values and determining the settings in which these conditions hold true.Throughout this exercise:

• You're tasked with identifying which \( t \) values satisfy conditions such as \( h(?) = h(-2) \).
• It requires you to know not just the value \( h(-2) = 0 \) but also to find what other \( t \) value possesses an equivalent output, which might mean such direct comparison simplifies this to \( h(-2) = h(-2) \).
• Recognizing unmatched circumstances, like in part d, where \( h(1) + h(2) eq h(t) \) for any \( t \), is also part of function identification.

Effectively identifying functions lets you discern patterns and apply known methods to solve new problems.