Problem 20 - Algebra Form and Function

Question

Evaluate the expressions given that $$ h(t)=10-3 t $$ $$ h\left(4-t^{3}\right) $$

Step-by-Step Solution

Answer

Answer: The simplified expression for h(4 - t^3) is 3t^3 - 2.

Step 1: Identify the function

The function is h(t) = 10 - 3t.

Step 2: Substitute the given input into the function

We are given the input (4 - t^3). So, we will substitute that into the function h(t) to find the value of h(4 - t^3): $$ h(4 - t^3) = 10 - 3(4 - t^3) $$

Step 3: Simplify the expression

Distribute the 3 and simplify the expression: $$ h(4 - t^3) = 10 - 12 + 3t^3 $$ $$ h(4 - t^3) = -2 + 3t^3 $$ Now the expression is evaluated and simplified to: $$ h(4 - t^3) = 3t^3 - 2 $$

Key Concepts

Substitution

When dealing with function evaluation, substitution is a crucial step where we replace a variable with a specific value or expression. This allows us to evaluate the function at that particular point or scenario.
Consider the function given in the original exercise, \( h(t) = 10 - 3t \). In this case, we need to evaluate this function for a specific input, \( 4 - t^3 \).

• Substitution involves replacing the original variable, in this case \( t \), with the new given expression \( 4 - t^3 \).


• This leads to the modified function: \( h(4 - t^3) = 10 - 3(4 - t^3) \).

Substitution is sometimes compared to filling in the blanks, where the blanks represent the variable spots in the function that need to be filled by the given expression or value. This allows you to tailor the function to match specific scenarios, essentially shifting the function's focus based on your substitution.

Simplification

After substitution, the next step is simplification, which involves making the expression easier to work with. The goal of simplification is to reduce an expression to its simplest form, ensuring clarity and ease of use for subsequent calculations.
For our exercise, once we've substituted \( t \) with \( 4 - t^3 \), we have the expression \( h(4 - t^3) = 10 - 3(4 - t^3) \). Here's how we go about simplifying it:

• Distribute the multiplication of 3 across the expression \( (4 - t^3) \), resulting in \( -12 + 3t^3 \).


• Next, combine any like terms. Here, combine the constant terms: \( 10 - 12 \) gives us \( -2 \).

The simplified version of the expression is \( 3t^3 - 2 \). Simplification is indispensable for making expressions readily interpretable and efficient for further computations. It strips down an expression to its core operational components.

Expression Evaluation

The final phase in evaluating a function expression involves pulling all your work together to confirm you have the simplest and most useful expression. Expression evaluation is the act of calculating the value of a function for a particular variable input, once all substitutions and simplifications have been completed.
Given the simplified expression from the previous step, \( h(4 - t^3) = 3t^3 - 2 \), we now understand the final form of the function for this specific input. You're essentially checking:

• The variable substitutions are correctly implemented.


• Simplifications are as minimal as possible.


• The resulting expression accurately represents the behavior of the function for the given input.

This phase of expression evaluation can also include calculating specific values if \( t \) were to be replaced by a numerical value. But in our case, we've set up a general solution as \( 3t^3 - 2 \), which can be directly used for further evaluations or analysis.