The table below gives the values of 2 functions, \(f\) and \(g,\) for various values of \(x\) . One of the functions. expresses a linear relationship. What is the value of that function at \(x=4\) ? \(\begin{array}{|c|c|c|}\hline x & {f(x)} & {g(x)} \\ \hline-2 & {1.4} & {0.6} \\\ {-1} & {1.2} & {0.6} \\ {0} & {} & {} \\ {2} & {0.8} & {1.3} \\ {3} & {0.6} & {1.6} \\ {4} & {} \\ \hline\end{array}\) A. 0.2 B. 0.4 C. 1.9 D. 2.0 E. 2.2

When we talk about function values, we're referring to the outputs that correspond to specific inputs of a function. In algebra, functions are often displayed in a table that shows the input values (usually represented as 'x'), and their corresponding function values (the outputs, such as 'f(x)' or 'g(x)'). Understanding function values is critical; they represent the essence of a function's behavior and allow us to interpret and predict the function's pattern.

For instance, in a linear function, as the input value increases or decreases by a constant amount, the function values also change by a constant rate. This constant rate of change is a key aspect in identifying a linear relationship. It's the steady, predictable alteration of values that distinguishes linear functions from other types of functions, like quadratic or exponential functions, which have variable rates of change. The ability to interpret these function values is important for solving problems involving linear equations and for understanding the concept of the slope in a graphical representation.

Linear relationships are a fundamental concept in algebra. To identify whether a function represents a linear relationship, we look for uniform changes in the function values over equal intervals of the input variable. This relationship is commonly visualized on a graph as a straight line, which is why we call it 'linear'—straight across the plane.

How do we identify linear relationships in a table of values or when analyzing an equation? It's simple: we need to determine whether there is a constant rate of change, also known as the slope. For every unit increase or decrease in the input value, if the output value changes by a consistent amount, the function is linear. This is the hallmark of a linear relationship. In contrast, if the rate of change varies, then the relationship is not linear—it might be quadratic, exponential, or follow some other pattern. Recognizing linear relationships is crucial for graphing straight lines and for understanding various mathematical models that rely on proportionality and direct variation.

The concept of a constant difference is particularly significant when working with linear functions. A 'constant difference' means that as we move from one input value to the next, the change in the function values is always the same. This is a key factor in identifying linear functions among various types of relations.

A straightforward method to verify a constant difference is to take any two successive pairs of function values and subtract them. If the function is linear, the differences will be equal. For example, if we subtract the output of a function at 'x = 1' from the output at 'x = 2', and do the same for 'x = 2' and 'x = 3', both differences should be identical. This property makes it easier for anyone trying to identify linear functions based on a given set of data points—as seen in our original problem referencing 'f(x)' and 'g(x)'. Also, understanding the idea of constant difference can simplify the process of predicting future values of the function, making it a powerful tool for solving algebraic problems and analyzing real-world situations where linear models are applicable.