Problem 54 - College Algebra

Question

f and g are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{cc}x & f(x) \\ \hline-1 & 1 \\ 0 & 4 \\ 1 & 5 \\ 2 & -1 \end{array}$$ $$\begin{array}{cc}x & g(x) \\ \hline-1 & 0 \\ 1 & 1 \\ 4 & 2 \\ 10 & -1 \end{array}$$ $$ f(g(4)) $$

Step-by-Step Solution

Answer

-1

Step 1: Evaluate \(g(4)\)

First, find the value of \(g(x)\) when \(x = 4\). According to the table, \(g(4)=2\).

Step 2: Substitute \(g(4)\) into \(f(x)\)

Next, substitute \(g(4)\) equals 2 into the function \(f\). We need to find \(f(g(4))\) which becomes \(f(2)\) after substitution.

Step 3: Evaluate \(f(2)\)

Finally, find the value of \(f(x)\) when \(x = 2\). According to the function f's table, \(f(2) = -1\).

Key Concepts

Function Composition

Function composition is a fundamental concept in mathematics that involves applying one function to the results of another. It's written as f(g(x)) and is read as 'f of g of x.' The process essentially feeds the output of one function directly into the input of another.

To effectively understand function composition, imagine a scenario where you have two machines. The first machine takes an object and applies a certain transformation to it, and then the output is immediately put into a second machine, which also transforms it. The overall effect is a combination of two processes.

With function composition, we follow a specific order:

• Evaluate the innermost function first. In the example f(g(4)), we begin by finding g(4).
• Use the result from the first step as the input for the next function. So after determining g(4), we then evaluate f(g(4)), which is the same as f(2) after substituting the obtained value.

Carefully evaluating each function step by step allows for accurate composition of functions.

Evaluating Functions

Evaluating functions is a quintessential skill in mathematics, allowing students to determine the output of a function given a specific input. When confronted with a function table, the task is straightforward: locate the input along one axis and find the corresponding output on another.

This direct method of retrieval helps in understanding how a function behaves at certain points and forms the basis for further mathematical analysis, such as graphing or solving equations. In our problem, evaluating g(4) requires us to find the output when x = 4, which is 2 according to the table. Following this, we use this output as the new input for another function evaluation, f(2), to complete the function composition process.

Piecewise-defined Functions

Piecewise-defined functions are essentially functions that have different expressions or rules for different intervals of the input variable. These are particularly useful when the relationship between the independent and dependent variables changes based on the input's value.

While not explicitly featured in the example problem, understanding piecewise functions aids in making sense of why a function might be represented in a table form. Each row in the table can be thought of as a piece of the function with its own rule. These pieces come together to define the behavior of the function across its entire domain.

Tables, such as the ones in our problem, can represent simplified piecewise functions where each input has a distinctly defined output, emphasizing the idea that a function may act differently across various segments of its domain.