Function Values

As we have seen in previous chapters, to evaluate a function, we substitute a value for the independent variable in the function equation and simplify the arithmetic expression. This value is the function value corresponding to the value of the independent variable. For example, to evaluate the value of the function [latex]f(x)=x^3-2x^2+5x-14[/latex] when [latex]x = 1[/latex], we substitute [latex]x[/latex] with the value 1 and simplify:

 [latex]\begin{aligned} f(x)&=x^3-2x^2+5x-14\\f(1) &= (1)^3 - 2(1)^2 + 5(1) - 14 \\f(1)&=1-2+5-14 \\f(1)&= -10\end{aligned}[/latex]

Therefore, the function value when [latex]x = 1[/latex] is –10, or, [latex]f(1) = -10[/latex].

The Meaning of Assigning a Function a Value

Assigning a polynomial function a value (e.g., [latex]f(x) = 2[/latex]) has several meanings depending on perspective. First, it means finding the value(s) in the domain that is (are) mapped onto this function value in the range. For example, figure 1 shows the mapping of [latex]x[/latex]-values to [latex]f(x)=x^2[/latex]. If [latex]f(x)=a[/latex], we need to find the value in the domain that results in a value of [latex]a[/latex] for [latex]f(x)[/latex].

Figure 1.The meaning of assigning a function a value: mapping

Second, since a function is composed of ordered pairs [latex](x, y)[/latex] (or points [latex](x, y)[/latex] on the coordinate plane), it may be understood as finding the [latex]x[/latex]-coordinate given the value of the [latex]y[/latex]-coordinate on the coordinate plane. For example, figure 2 shows the graph of a polynomial [latex]y=f(x)[/latex]. If [latex]f(x)=1[/latex] then [latex]y=1[/latex] and the corresponding points on the graph when [latex]y=1[/latex] are (–3, 1) and (–1, 1). Consequently, when [latex]f(x)=1[/latex], [latex]x=–1,\;–3[/latex].

FIgure 2. The meaning of assigning a function a value: coordinate points on a graph

Third, assigning a function a value, [latex]f(x) = 2[/latex] for example, means finding the intersection point between the graph of the function and the horizontal line [latex]y = 2[/latex]. Figure 3 shows the function [latex]f(x)=x^2-4x+5[/latex] and the horizontal line [latex]y=2[/latex]. The parabola and the line intersect at the points (1, 2) and (3, 2). This shows that when [latex]f(x)=2[/latex], the corresponding [latex]x[/latex]-values are 1 and 3.

Figure 3. The meaning of assigning a function a value: intersection of graph with horizontal line

Addition and Subtraction

We can add and subtract polynomial functions by combining like terms. For example, [latex]5{x}^{2}[/latex] and [latex]-2{x}^{2}[/latex] are like terms and can be added to get [latex]3{x}^{2}[/latex], but [latex]3x[/latex] and [latex]3{x}^{2}[/latex] are not like terms, since they have different exponents, therefore, they cannot be added.

Another way to organize terms is do a hybrid version of the horizontal and vertical methods by writing like terms beneath one another as we distribute horizontally. This collects like terms in the same column so they are easily recognized and combined.

We can multiply polynomial functions using the distributive property no matter how many terms are in the polynomials.