5 BASIC GRAPHSTHE FOLLOWING ARE THE GRAPHS that occur throughout analytic geometry and calculus. The student should be able to sketch them -- and recognize them -- purely from their shape. It is not necessary to plot points. A constant function Here is the graph of y = f(x) = 3. It is a straight line parallel to the x-axis. It is called a constant function because to every value of x there corresponds the same value of y: 3. Is a constant function single-valued? Yes, it is, because to each value of x there is one and only one value of y. 3. A constant function has the form y = c , where c is a constant, that is, a number. The identity function and the absolute value function y = x is called the identity function because the value of y is identical with that of x. The coördinate pairs are (x, x). In the absolute value function, the negative values of y in the identity function are reflected into the positive side. For, |−x| = |x| = x. The coördinate pairs are (x, |x|). Example. a) What is the domain of the identity function? There is no natural restriction on the values of x. Therefore, the domain -- where the function "lives" -- includes every real number. − Note first that infinity " Note that we write "x less than b) What is the range of the identity function? The range are those values of y that correspond to the values in the domain. Inspecting the graph will show that y, also, will take every real value. − Parabola and square root function In the parabola y = x2, the coördinate pairs are (x, x2). We can see that the following points are on the graph: (1, 1), (−1, 1), (2, 4), (−2, 4), and so on. The graph of the square root function is related to y = x2. It is its inverse. The coördinate pairs are (x, Note that the square root function is defined only for non-negative values of x. For, the square root of a negative number is not real. Also, the symbol Problem 1. What is the domain of the function y =x2, and what is its range? This function is defined for all values of x : −∞ < x < ∞. As for the range, the lowest value of y is 0. And there is no limit to the highest value. 0 ≤ y < ∞. Problem 2. What is the domain of the square root function, and what is its range? The square root function is defined only for non-negative values of x. Domain: x ≥ 0. As for the range, the lowest value of y is 0. And there is no limit to the highest value. 0 ≤ y < ∞. The cubic function The cubic function is y = x3. When x is negative, y is negative: Odd powers of a negative number are negative. Problem 3. What is the domain of the cubic function, and what is its range? Domain: −∞ < x < ∞. Range: −∞ < y < ∞. The reciprocal function When x is a very large positive number -- on the extreme right of the x-axis -- its reciprocal is a very small positive number. The graph is very close to the x-axis. When x is a very small positive number -- close to x = 0 -- its reciprocal is a very large positive number. Similar properties hold when x is negative. Note, however, that x may not be 0. 0 is the only value that must be excluded from the domain. We will go into this more in Topic 18. ![]() Next Topic: The vocabulary of polynomial functions Please make a donation to keep TheMathPage online. Copyright © 2018 Lawrence Spector Questions or comments? E-mail: themathpage@yandex.com Private tutoring available. |