Problem 17 - Algebra Form and Function

Question

Evaluate the expressions given that $$ h(t)=10-3 t $$ $$ h(k-3) $$

Step-by-Step Solution

Answer

Answer: The expression for \(h(k-3)\) is \(-3k + 19\).

Step 1: Write the given function and expression

We have the function: $$ h(t) = 10 - 3t $$ And we are asked to find: $$ h(k-3) $$

Step 2: Substitute \((k-3)\) for \(t\) in the equation

Replace \(t\) with \((k-3)\) in the function \(h(t)\): $$ h(k-3) = 10 - 3(k-3) $$

Step 3: Simplify the expression

Distributing the \(-3\) term: $$ h(k-3) = 10 - 3k + 9 $$ Combine the constants: $$ h(k-3) = -3k + 19 $$

Step 4: State the final expression

The expression \(h(k-3)\) is simplified to: $$ h(k-3) = -3k + 19 $$

Key Concepts

Function Notation

Function notation is a way to represent functions in a concise manner. It uses a specific format that involves the name of the function, usually labeled such as \( f(x) \), \( g(x) \), or \( h(t) \), and it often includes the variable in parentheses. In our problem, the function is given as \( h(t) = 10 - 3t \). Here, \( h \) is the function name, and \( t \) is the variable.

Using this notation, we can easily express what happens to the original function when we modify or evaluate it with different inputs. The main advantage is that it allows us to see at a glance what variable affects the function and how any particular input will transform the function's output. This notation is efficient and helps in clearly understanding the relationship between variables and the operations applied to them.

Function notation is an essential building block in algebra and higher mathematics, as it simplifies the evaluation of functions with different inputs and helps keep equations organized.

Substitution in Functions

Substitution is like a "plug-in" process where we replace the variables in a function with specific values or expressions. With substitution, we can evaluate how different inputs affect the function. In our case, to find \( h(k-3) \), we substitute \((k-3)\) for \(t\) in the function \( h(t) = 10 - 3t \).

Here's how it works:

• Start with the given function: \( h(t) = 10 - 3t \)
• Replace \( t \) with \( k-3 \): \( h(k-3) = 10 - 3(k-3) \)

Substitution transforms the general expression into a specific one based on the inserted values. This technique is crucial for evaluating functions, solving equations, and understanding how function outputs change with different inputs. It is a fundamental concept for problem-solving in algebra.

Algebraic Simplification

Algebraic simplification involves making an expression easier to read and work with by combining like terms and simplifying expressions. After substitution, the next step is to simplify the expression we obtained. Let's simplify \( h(k-3) = 10 - 3(k-3) \):

1. First, distribute the \(-3\) to both terms inside the parentheses:

• \( h(k-3) = 10 - 3k + 9 \)

2. Then, combine the constant terms:

• Combine \(10\) and \(9\) to get \(19\).
• So, \( h(k-3) = -3k + 19 \)

Algebraic simplification helps in reducing complexity for easier interpretation and calculation. It often reveals the core meaning or value of the expression, making the solution clear and straightforward. Mastery of simplification allows for proper manipulations necessary to efficiently solve equations and analyze functions in mathematics.