Problem 36 - Elementary and Intermediate Algebra

Question

Let \(s(x)=3-x\) and \(t(x)=x^{2}-x-6 .\) Find each function value. $$ (s-t)(12) $$

Step-by-Step Solution

Answer

(s-t)(12) = -135.

Step 1: Understand the Composite Function

We need to find the value of the function \((s-t)(x)\) at \(x = 12\). The function \((s-t)(x)\) is defined as \(s(x) - t(x)\).

Step 2: Write the Expression for (s-t)(x)

We have two functions, \(s(x) = 3 - x\) and \(t(x) = x^2 - x - 6\). Therefore, the expression for \((s-t)(x)\) will be \((s-t)(x) = s(x) - t(x) = (3-x) - (x^2-x-6)\).

Step 3: Simplify the Expression

Simplify the expression: \((3-x) - (x^2-x-6) = 3 - x - x^2 + x + 6\). Combine like terms: \(= -x^2 + (x-x) + (3+6) = -x^2 + 9\). Therefore, \((s-t)(x) = -x^2 + 9\).

Step 4: Substitute x with 12

Now substitute 12 into the expression for \((s-t)(x)\): \((s-t)(12) = -12^2 + 9\).

Step 5: Calculate the Expression

Calculate \(-12^2 + 9\). First, \(12^2 = 144\), so \(-12^2 = -144\). Then, \(-144 + 9 = -135\).

Step 6: Final Answer

The value of \((s-t)(12)\) is \(-135\).

Key Concepts

Composite Function

A composite function in algebra takes two functions and combines them using operations like addition, subtraction, multiplication, or division. In our given problem, we combined two functions: \( s(x) = 3 - x \) and \( t(x) = x^2 - x - 6 \). The operation used here is subtraction, forming a new function \((s-t)(x) = s(x) - t(x)\). By creating this composite, you bring the two functions together into a single expression. This allows us to analyze and work with them as one, instead of separately. It's important to master composite functions because they are a key concept in higher-level math topics like calculus. When dealing with composite functions, always remember to pay close attention to the operations involved and the order in which they occur.

Substitution Method

The substitution method is crucial when working with composite functions in algebra. It's a technique where we replace a variable with a specific value or another expression to simplify the problem or find a specific result. In this exercise, the task was to find \((s-t)(12)\), which means replacing \(x\) with 12 in our composite function. To correctly apply the substitution method, follow these tips:

• Identify the variable you need to substitute and where it appears in your expression.
• Replace the variable with the given number or expression, maintaining the correct order of operations.
• Double-check your results to ensure accuracy.

This method is very effective for evaluating expressions at specific points, giving you concrete numbers instead of variable outcomes.

Simplification

Simplification is the process of making an algebraic expression easier to read or solve. This involves combining like terms, removing parentheses, and restructuring terms to reach the simplest form. In our problem, after creating our composite function \((s-t)(x) = (3-x) - (x^2-x-6)\), we simplified:

• Expand all parts of the expression and remove parentheses.
• Combine like terms, organizing them from highest to lowest degree.
• The expression simplifies to \(-x^2 + 9\).

Streamlining expressions like this makes further algebraic manipulation easier and reduces the chances of errors in calculations. This skill is essential for successfully navigating complex algebraic operations.

Expression Evaluation

Once you've simplified your expression, it's time to evaluate it at the given point. In this case, we want to find the value at \(x = 12\) for our simplified expression \(-x^2 + 9\).Steps involved in expression evaluation:

• Replace \(x\) with 12 in the expression \(-x^2 + 9\).
• Calculate \(-12^2\), which is \(-144\).
• Then add 9 to \(-144\), obtaining \(-135\).

Evaluating expressions accurately is critical, as it provides us with specific numerical results. In this exercise, the evaluation shows the final answer: \( (s-t)(12) = -135 \). Understanding this process strengthens problem-solving skills and ensures precision in algebra.