The algebraic way see if an equation determines y as a function of x is to solve for y. If there is not a unique solution, then y is not a function of x. Suppose that we are given the graph of the equation. There is an easy way to see if this equation describes y as a function of x. A set of points in the plane is the graph of a function if and only if no vertical line intersects the graph in more than one point.
By the vertical line test, this graph is not the graph of a function, because there are many vertical lines that hit it more than once. Think of the vertical line test this way. The points on the graph of a function f have the form x, f x , so once you know the first coordinate, the second is determined. Therefore, there cannot be two points on the graph of a function with the same first coordinate.
All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph twice, then there are two points on the graph with the same first coordinate. If that happens, the graph is not the graph of a function. Think of a point moving on the graph of f. As the point moves toward the right it rises. This is what it means for a function to be increasing.
Your text has a more precise definition, but this is the basic idea. The function f above is increasing everywhere. In general, there are intervals where a function is increasing and intervals where it is decreasing. The function graphed above is decreasing for x between -3 and 2. It is increasing for x less than -3 and for x greater than 2. Some of the most characteristics of a function are its Relative Extreme Values. Points on the functions graph corresponding to relative extreme values are turning points, or points where the function changes from decreasing to increasing or vice versa.
Let f be the function whose graph is drawn below. Note that f a is not the smallest function value, f c is. However, if we consider only the portion of the graph in the circle above a, then f a is the smallest second coordinate. Look at the circle on the graph above b. While f b is not the largest function value this function does not have a largest value , if we look only at the portion of the graph in the circle, then the point b, f b is above all the other points.
So, f b is a relative maximum of f. Indeed, f c is the absolute minimum of f, but it is also one of the relative minima. Once we have determined that a graph defines a function, an easy way to determine if it is a one-to-one function is to use the horizontal line test. Draw horizontal lines through the graph.
The function in a is not one-to-one. The horizontal line shown below intersects the graph of the function at two points and we can even find horizontal lines that intersect it at three points. The function in b is one-to-one. Any horizontal line will intersect a diagonal line at most once.
In this text we explore functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers.
When working with functions, it is similarly helpful to have a base set of building-block elements. Some of these functions are programmed to individual buttons on many calculators. We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties.
The graphs and sample table values are included with each function shown below. Improve this page Learn More. Skip to main content. Determine the domain and range of the cube root function.
Find the ordered pair that specifies the point P. Graph the piecewise functions. Evaluate given the graph of f. The value of an automobile in dollars is given in terms of the number of years since it was purchased new in The cost per unit in dollars of custom lamps depends on the number of units produced according to the following graph:.
Graph the cost of the rental boat and determine the cost to rent the boat for 4 1 2 hours. Explain to a beginning algebra student what an asymptote is. Research and discuss the difference between the floor and ceiling functions. What applications can you find that use these functions? Previous Section. Table of Contents.
Next Section. Define and graph piecewise functions. Evaluate piecewise defined functions. Define the greatest integer function. Basic Functions In this section we graph seven basic functions that will be used throughout this course. Piecewise Defined Functions A piecewise function A function whose definition changes depending on the values in the domain. Solution: In this case, we graph the squaring function over negative x -values and the square root function over positive x -values.
Key Takeaways Plot points to determine the general shape of the basic functions. The shape, as well as the domain and range, of each should be memorized.
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