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Find the domain and range for $f\left(x\right)=\sqrt{4-2x}+5.$
Domain = $\left\{x|x\le 2\right\},$ range = $\left\{y\right|y\ge 5\}$
Typically, a function is represented using one or more of the following tools:
We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.
Functions described using a table of values arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable $x$ be the time after midnight, measured in hours, and the output variable $y$ be the temperature $x$ hours after midnight, measured in degrees Fahrenheit. We record our data in [link] .
Hours after Midnight | Temperature $(\text{\xb0}F)$ | Hours after Midnight | Temperature $(\text{\xb0}F)$ |
---|---|---|---|
0 | 58 | 12 | 84 |
1 | 54 | 13 | 85 |
2 | 53 | 14 | 85 |
3 | 52 | 15 | 83 |
4 | 52 | 16 | 82 |
5 | 55 | 17 | 80 |
6 | 60 | 18 | 77 |
7 | 64 | 19 | 74 |
8 | 72 | 20 | 69 |
9 | 75 | 21 | 65 |
10 | 78 | 22 | 60 |
11 | 80 | 23 | 58 |
We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.
Given a function $f$ described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in [link] can give us a better idea of their fluctuation throughout the day. [link] shows the plot of the temperature function.
From the points plotted on the graph in [link] , we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in [link] .
Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius $r$ is given by the formula $A\left(r\right)=\pi {r}^{2}.$ When an object is thrown upward from the ground with an initial velocity ${v}_{0}$ ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula $s\left(t\right)=\mathrm{-16}{t}^{2}+{v}_{0}t.$ When $P$ dollars are invested in an account at an annual interest rate $r$ compounded continuously, the amount of money after $t$ years is given by the formula $A\left(t\right)=P{e}^{rt}.$ Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.
Given an algebraic formula for a function $f,$ the graph of $f$ is the set of points $\left(x,f\left(x\right)\right),$ where $x$ is in the domain of $f$ and $f(x)$ is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of $f$ consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.
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