## 1.2 Functions from the Graphical Viewpoint

(This topic is also in Section 1.2 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)

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Let us start by looking at the definition in the textbook (and also in the Chapter Summary)

Graph of a Function

The graph of the function f is the set of all points (x, f(x)) in the xy-plane, where we restrict the values of x to lie in the domain of f.

To obtain the graph of a function, plot points of the form (x, f(x)) for several values of x in the domain of f. The shape of the entire graph can usually be inferred from sufficiently many points.

Example

To sketch the graph of the function

 f(x) = x2 Function notation y = x2 Equation notation
with domain the set of all real numbers, first choose some values of x in the domain and compute the corresponding y-coordinates.
 x -3 -2 -1 0 1 2 3 y = x2 9 4 1 0 1 4 9

Plotting these points gives the picture on the left, suggesting the graph on the right

If only the graph of a function is given to begin with, we say that the function has been specified graphically. Here is an example of a graphically specified function.

The following graph shows the total population in state and federal prisons in 1970-1997 as a function of time in years (t = 0 represents 1970).*

* Data are approximate. Sources: Bureau of Justice Statistics, New York State Dept. of Correctional Services/The New York Times, January 9, 2000, p. WK3.

 P(25) - P(20) =

Q The above answer tells us that the number of prisoners

 Select one was 400,000 by 1995 was 400,000 in 1990 increased by 400,000 in each of 1980 and 1982 increased by 40% over the period 1990-1995 increased by 400,000 over the period 1990-1995 increased by 40% per year over the period 1990-1995 did not change significantly over the period 1990-1995

The next example shows that plotting a few points might not convey anough infomration to enable us to draw a graph.

 Let f(x) = x + 1x . Sketch its graph.

To obtain its graph, we must first sompute some values of f(x). Enter the missing values in the given cells. (Use improper fraction notation, like "-7/4" or decimals accurate to 4 places to fill in the cells. NOTE: If your decimals are not accurate to 4 places, your answer will be marked wrong.)

x-3-2-10123
 y = x + 1x
Undefined

Here is what we get if we carefully plot the points we just obtained.

Now, which one is its graph? (Click on the correct graph, and press "Help" to see why it is correct.)

### The Graph of a Piecewise Defined Function

Since piecewise-defined functions are based on more than one-formula, their graphs are composed of more than one curve. Here is Example 4 in the textbook: Let

f(x)=  -1 if -4 x < -1 x if -1 x 1 x2-1 if 1 < x 2

The graph of f consists of portions of three graphs superimposed. To see how they fit together, click the buttons under the graph of f below.

Now try some of the exercises in Section 1.2 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.

Last Updated: March, 2006