Subsection5.2.1Reading Function Values from a Graph
The graph below shows the Dow-Jones Industrial Average (the average value of the stock prices of 500 major companies) recorded at noon each day during the stock market correction around October 10, 1987 ("Black Monday").
The graph describes a function because there is only one value of the output, DJIA, for each value of the input, \(t\text{.}\) There is no formula that gives the DJIA for a particular day; but it is still a function, defined by its graph. The value of \(f(t)\) is specified by the vertical coordinate of the point with the given \(t\)-coordinate.
We can say that \(f(20) = 1726\text{,}\) so the point \((20, 1726)\) lies on the graph of \(f\text{.}\) This point is labeled \(Q\) in the figure above.
The water level in Lake Huron alters unpredictably over time. The graph below gives the average water level, \(L=f(t)\text{,}\) in meters in the year \(t\) over a 20-year period. (Source: The Canadian Hydrographic Service)
To find \(g(-2)\text{,}\) we look for the point with \(t\)-coordinate \(-2\text{.}\) The point \((-2, 0)\) lies on the graph of \(g\text{,}\) so \(g(-2) = 0\text{.}\) Similarly, the point \((5, 1)\) lies on the graph, so \(g(5) = 1\text{.}\)
We look for points on the graph with \(y\)-coordinate \(-2\text{.}\) Because the points \((-5, -2)\text{,}\)\((-3, -2)\text{,}\) and \((3, -2)\) lie on the graph, we know that \(g(-5) = -2\text{,}\)\(g(-3) = -2\text{,}\) and \(g(3) = -2\text{.}\) Thus, the \(t\)-values we want are \(-5\text{,}\)\(-3\text{,}\) and \(3\text{.}\)
The highest point on the graph is \((1, 4)\text{,}\) so the largest \(y\)-value is \(4\text{.}\) Thus, the maximum value of \(g(t)\) is \(4\text{,}\) and it occurs when \(t = 1\text{.}\)
A graph is increasing if the \(y\)-values get larger as we read from left to right. The graph of \(g\) is increasing for \(t\)-values between \(-4\) and \(1\text{,}\) and between \(3\) and \(5\text{.}\) Thus, \(g\) is increasing on the intervals \((-4, 1)\) and \((3, 5)\text{.}\)
Subsection5.2.2Constructing the Graph of a Function
We can construct a graph for a function described by a table or an equation. We make these graphs the same way we graph equations in two variables: by plotting points whose coordinates satisfy the equation.
We choose several convenient values for \(x\) and evaluate the function to find the corresponding \(f(x)\)-values. For this function we cannot choose \(x\)-values less than \(-4\text{,}\) because the square root of a negative number is not a real number.
Points on the graph have coordinates \((x, f(x))\text{,}\) so the vertical coordinate of each point is given by the value of \(f(x)\text{.}\) We plot the points and connect them with a smooth curve, as shown in the figure. Notice that no points on the graph have \(x\)-coordinates less than \(-4\text{.}\)
Technology5.2.9.Using Technology to Graph a Function.
We can also use a graphing utility to obtain a table and graph for the function in ExampleΒ 5.2.7. We graph a function just as we graphed an equation. For this function, we enter
and press ZOOM\(6\) for the standard window. Your calculator does not use the \(f(x)\) notation for graphs, so we will continue to use \(Y_1,~ Y_2,\) etc. for the output variable. Donβt forget to enclose \(x+4\) in parentheses, because it appears under a radical. The graph is shown below.
In a function, two different outputs cannot be related to the same input. This restriction means that two different ordered pairs cannot have the same first coordinate. What does it mean for the graph of the function?
Consider the graph shown in Figure (a). Every vertical line intersects the graph in at most one point, so there is only one point on the graph for each \(x\)-value. This graph represents a function.
In Figure (b), however, the line \(x=2\) intersects the graph at two points, \((2,1)\) and \((2,4)\text{.}\) Two different \(y\)-values, 1 and 4, are related to the same \(x\)-value, 2. This graph cannot be the graph of a function.
For graph (c), notice the break in the curve at \(x = 2\text{:}\) The solid dot at \((2, 1)\) is the only point on the graph with \(x = 2\text{;}\) the open circle at \((2, 3)\) indicates that \((2, 3)\) is not a point on the graph. Thus, graph (c) is a function, with \(f(2) = 1\text{.}\)
Subsection5.2.4Graphical Solution of Equations and Inequalities
We have used graphs to solve linear and quadratic equations and inequalities. We can also use a graphical technique to solve equations and inequalities involving other functions.
If we sketch in the horizontal line \(y = 15\text{,}\) we can see that there are three points on the graph of \(f\) that have \(y\)-coordinate \(15\text{,}\) as shown below. The \(x\)-coordinates of these points are the solutions of the equation.
From the graph, we see that the solutions are \(x = -3\text{,}\)\(x = 1\text{,}\) and approximately \(x = 2.5\text{.}\) We can verify each solution algebraically.
We first locate all points on the graph that have \(y\)-coordinates greater than or equal to \(15\text{.}\) The \(x\)-coordinates of these points are the solutions of the inequality.
The figure below shows the points in red, and their \(x\)-coordinates as intervals on the \(x\)-axis. The solutions are \(x \le -3\) and \(1\le x \le 2.5\text{,}\) or in interval notation, \((-\infty, -3] \cup [1, 2.5]\text{.}\)
To simplify the notation, we sometimes use the same letter for the output variable and for the name of the function. In the next example, \(C\) is used in this way.
TrailGear decides to market a line of backpacks. The cost, \(C\text{,}\) of manufacturing backpacks is a function of the number of backpacks produced, \(x\text{,}\) given by the equation
\begin{equation*}
C=C(x)=3000+20x
\end{equation*}
where \(C(x)\) is in dollars. Find the cost of producing 500 backpacks.
The graph shows the speed of sound in the ocean as a function of depth, \(S = f (d)\text{.}\) The speed of sound is affected both by increasing water pressure and by dropping temperature. (Source: Scientific American)
The figure shows the graph of \(g(x) = \dfrac{12}{2 + x^2}\text{.}\) Use the graph to solve the following equations and inequalities. Show your work on the graph. Write your answers in interval notation.
The figure shows the graph of \(H(t)=t^3-4t^2-4t+12\text{.}\) Use the graph to solve the following equations and inequalities. Show your work on the graph. Write your answers to the inequalities in interval notation.