Problem 16 - Contemporary Precalculus

Question

Find the indicated values, where $$ g(t)=t^{2}-t \text { and } f(x)=1+x B$$ $$f(2 g(1))$$

Step-by-Step Solution

Answer

Answer: The value of \(f(2g(1))\) is 1.

Step 1: Calculate \(g(1)\)

To find the value of \(g(1)\), substitute \(t = 1\) into the given \(g(t)\) equation: $$g(t) = t^2 - t$$ $$g(1) = 1^2 - 1 = 1 - 1 = 0$$ So, \(g(1) = 0\).

Step 2: Calculate \(2g(1)\)

Now that we have the value for \(g(1)\), we can calculate \(2g(1)\): $$2g(1)=2 \times 0 = 0$$

Step 3: Calculate \(f(2g(1))\)

Substitute the value of \(2g(1)\) into the given \(f(x)\) equation: $$f(x) = 1 + xB$$ $$f(2g(1)) = 1 + (0)B = 1$$ So, the value of \(f(2g(1))=1\).

Key Concepts

Function Composition

Function composition involves combining two functions, such that the output of one function becomes the input of another. This process can help solve complex problems by breaking them into simpler parts. In the context of our exercise, we perform function composition by substituting the output of the function \( g(t) \) into the function \( f(x) \).

When composing functions, consider the following steps:

• Identify the inner function and determine its output based on the input value.
• Use the output of the inner function as the input for the outer function.
• Calculate the final result from the outer function using the given formula.

Here, \( g(1) \) serves as the inner function positioned within \( f(x) \). We first find \( g(1) \), then set \( x = 2g(1) \) to calculate \( f(2g(1)) \). This structured approach simplifies the problem-solving process.

Algebraic Manipulation

Algebraic manipulation is a technique that involves rearranging and simplifying expressions to solve equations easily. This skill is essential for evaluating functions or expressions quickly.

In this exercise, algebraic manipulation helps in several ways:

• Substituting values into equations to isolate variables and find specific outputs.
• Multiplying expressions, as shown in calculating \( 2g(1) \), to find intermediate results.
• Simplifying complex expressions once substitutions are made, as seen in simplifying \( f(2g(1)) \).

By applying these methods, we can reduce errors and efficiently find the correct solution. Mastery of algebraic concepts simplifies not only composition but also various types of mathematical problems.

Mathematical Problem Solving

Solving mathematical problems typically involves understanding the problem thoroughly, then applying appropriate mathematical techniques and knowledge to find a solution. Each step in problem-solving should bring you closer to the answer.

The general approach for problem solving in exercises like this includes:

• Reading the problem carefully to understand what is being asked.
• Breaking the problem into manageable chunks, using functions as smaller tasks.
• Sequentially solving each part, as demonstrated stepwise in this example, from finding \( g(1) \) to calculating \( f(2g(1)) \).
• Checking each step for accuracy to ensure the final solution holds true.

By engaging in this methodical approach, you gain confidence in problem-solving abilities and reinforce your understanding of precalculus concepts like function composition and algebraic manipulation.