Problem 70 - College Algebra

Question

For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$\begin{array}{||cc|c|c|c|c|c|c|c|} \hline {x} & {-3} & {-2} & {-1} & {0} & {1} & {2} & {3} \\ \hline {f(x)} & {11} & {9} & {7} & {5} & {3} & {1} & {-1} \\\ \hline {g(x)} & {-8} & {-3} & {0} & {1} & {0} & {-3} & {-8}\\\ \hline \end{array}$$ $$(g \circ g)(1)$$

Step-by-Step Solution

Answer

The expression \((g \circ g)(1)\) evaluates to 1.

Step 1: Identify Known Values

From the given table, identify the value of \( g(1) \). The table shows \( g(1) = 0 \).

Step 2: Apply the Composition

Now, compute \((g \circ g)(1)\) by evaluating \(g(g(1))\). Since \( g(1) = 0 \), we need to find \( g(0) \) from the table.

Step 3: Find the Inner Function Result

Consult the table for \( g(0) \). It lists \( g(0) = 1 \).

Step 4: Conclusion of Composition

Thus, \((g \circ g)(1) = g(0) = 1\).

Key Concepts

Function Evaluation

Function evaluation is a fundamental concept in mathematics that involves finding the output of a function given an input value. Evaluating a function means substituting the desired input value into the function and then performing the operations defined by the function to obtain an output. This concept is crucial because it forms the basis for understanding how functions work and how different inputs can produce different outputs.
For example, let's consider the function \( g(x) \) from the table. If you are asked to evaluate \( g(1) \), you check the table and see that when \( x = 1 \), \( g(x) = 0 \). This tells you that the function \( g \) maps the input 1 to the output 0.
Function evaluation is not limited to single-step computations. It can involve more complex operations like composition, where you evaluate a function multiple times through another function. This brings us to our next topic, algebraic functions and their operations.

Algebraic Functions

Algebraic functions are mathematical expressions that involve operations like addition, subtraction, multiplication, division, and taking roots, all involving variables. They come in many forms and complexities, serving as the building blocks for more advanced mathematical concepts.
In the context of function composition, algebraic functions allow you to evaluate expressions such as \((g \circ g)(1)\). Here, composition refers to computing one function at the result of another, effectively "chaining" them together.
To understand \((g \circ g)(1)\), you first evaluate \(g(1)\), as mentioned earlier, which gives you \(0\). Then you substitute this result back into the same function, \(g\), to find \(g(0)\), resulting in the final output of \(1\). This illustrating the role of algebraic manipulation in solving problems using function composition.

Math Table Interpretation

Math table interpretation is an essential skill for understanding how functions behave based on given values. Tables provide a way to summarize information about a function at specific points, allowing you to easily find the needed information for evaluation and composition.
When reading a table, such as the one given in the exercise, each row corresponds to different functions, and each column presents the output for specific inputs.

• Identify which function the row represents, for example, \( f(x) \) or \( g(x) \).
• Locate the column for the specific input value you are interested in.
• Read off the corresponding function value from the correct table entry.

In our exercise, when determining \(g(1)\), you find the row for \(g(x)\), then locate where \(x = 1\) in the table, which gives you \(g(1) = 0\). For the inner composition step where you find \(g(0)\), follow the same process to conclude that \(g(0) = 1\). This clear understanding from a table is integral to resolving complex math problems involving functions.