Problem 58 - College Algebra

Question

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 3 to evaluate each expression. $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\hline {x} & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline {f(x)} & {7} & {6} & {5} & {8} & {4} & {0} & {2} & {1} & {9} & {3} \\ \hline {g(x)} & {9} & {5} & {6} & {2} & {1} & {8} & {7} & {3} & {4} & {0}\\\ \hline \end{array}$$ $$f(g(8))$$

Step-by-Step Solution

Answer

\(f(g(8)) = 4\)

Step 1: Determine the Inner Function Value

We need to find the value of the inner function first, which is \(g(8)\). According to the table, \(g(8) = 4\).

Step 2: Use the Inner Function Value

Now that we know \(g(8) = 4\), we use this result to find \(f(4)\). According to the table, \(f(4) = 4\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics where we determine the output of a function for a specific input. - **Understanding Function Evaluation**: In our case, we have functions \( f(x) \) and \( g(x) \) with predefined values for specified inputs, listed in a table. - For example, the table tells us that when \( x = 8 \), the function \( g(x) \) outputs 4 (i.e., \( g(8) = 4 \)).- **Evaluating Functions Using Given Data**: To evaluate a function, especially with provided tabular data like this, find the input value on the table's horizontal axis, then look vertically to find the corresponding function output. - This approach is crucial for solving function composition problems, making it easier to "read off" values without performing more complex calculations.

Algebraic Expressions

Algebraic expressions represent numbers, variables, and operations in a succinct format. - **Components of Algebraic Expressions**: They combine different elements: - Numbers (constants) - Variables (like \( x \) or \( y \)) - Operations (addition, subtraction, multiplication, division)- **Interpreting Expressions in Function Composition**: For function composition, algebraic expressions involve evaluating one function's output as the input to another function. This process simplifies complex calculations by allowing you to break down the problem into manageable parts. - For instance, evaluating \( f(g(x)) \) involves finding \( g(x) \) first and then using this result in \( f(x) \). - In our exercise, we computed \( g(8) = 4 \) first before evaluating \( f(4) \). This highlights how interrelated operations in algebra can be evaluated step-by-step for clear understanding.

Step-by-Step Solution

Solving problems systematically using a step-by-step approach helps simplify complex expressions and avoids confusions.Here is a detailed breakdown of the solution process for identifying \( f(g(8)) \):1. **Finding the Inner Function Value**: - First, identify the inner function. Here, it's \( g(8) \). Use the table to find its value. - The table tells us \( g(8) = 4 \).2. **Substituting Into the Outer Function**: - Use the value from the inner function (4) as the input for the outer function \( f(x) \). - Find \( f(4) \) in the table, which gives us \( f(4) = 4 \).By clearly following each step, you ensure accuracy by focusing on each evaluation in stages. This method is effective in managing more advanced algebraic problems, making each part of the solution manageable and easy to follow.