Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

The value is 9.

Step 1: Evaluate Inside Function

First, identify the inside function, which is \(f(5)\). By consulting the table, we find that \(f(5) = 0\).

Step 2: Substitute Inside Function Result

Substitute the result from Step 1 into the outside function, \(g(x)\), to get \(g(f(5)) = g(0)\).

Step 3: Evaluate Outer Function

Now evaluate \(g(0)\) using the table. \(g(0) = 9\).

Key Concepts

Function Evaluation

When working with functions, function evaluation involves finding the output of a function for a specific input. Understanding this is crucial for tackling problems in mathematics, especially when dealing with composite functions like in the given exercise. Simply put, function evaluation asks us to insert a specific value into a function and calculate the result.

A specific value, called the input, is applied to the function, represented as a mathematical formula or table. For example, if you have a function like \( f(x) = x^2 \), evaluating \( f(3) \) means replacing \( x \) with 3:

• \( f(3) = 3^2 = 9 \)

To perform function evaluation in a composite function such as \( g(f(x)) \), you first need to find \( f(x) \) before you can find \( g(x) \) for your final result.

Here, recognizing the sequence is key:

• Identify the "inner" function \( f(x) \)
• Find its value
• Use this as input for the "outer" function \( g(x) \)

Applying this step-by-step approach can help untangle any complex-looking function composition.

Tables in Algebra

In algebra, tables are a handy tool to express functions, particularly when functions are defined for a set of integer values or discrete inputs.

Tables provide a clear visual on how each input map to their corresponding outputs. They're especially useful when dealing with problems that require evaluation of multiple function values without delving into a more complicated formula or graph.

In our task, Tables 3 shows inputs \( x \) from 0 to 9, and their respective outputs for functions \( f(x) \) and \( g(x) \). Here's how to interpret and utilize them effectively:

• Find the input value on the table's top row.
• Track what output it gives by choosing the corresponding function below it (either \( f(x) \) or \( g(x) \)).

Using tables allows a systematic approach, always following the horizontal and vertical coordinates to access function values. This not only simplifies function evaluation but also minimizes computational errors.

Step-by-Step Solution

In algebra and calculus, a step-by-step solution process helps break down a problem into manageable parts. This is immensely beneficial in exercises involving function composition, like solving for \( g(f(5)) \).

Step-by-step breakdown:

• Step 1: Evaluate the Inside Function
  Identify what needs to be calculated first, the inner function \( f(x) \). For the input \( x = 5 \), the table provides \( f(5) = 0 \).


• Step 2: Substitute the Result in the Outside Function
  Substitute \( f(5) \) into \( g(x) \), leading to finding \( g(0) \). This showcases function composition, especially the order of operations.


• Step 3: Evaluate the Outer Function
  Finally, look at the table for \( g(0) \), which gives an output of 9. This illustrates how combining functions works through substitution and re-evaluation.

Dissecting each part of a solution teaches us to approach more intricate problems systematically and logically, ensuring we reach the correct answer through careful consideration of each step.