![]() ![]() a few points (x and y coordinates for both “parts” of the reciprocal function)įor the basic reciprocal function f(x) = 1/x, the vertical asymptote is at x = 0.the horizontal asymptote (a line y = b that the function does not intersect.the vertical asymptote (a line x = a that the function does not intersect).To draw a reciprocal function, it helps to find a few key features first: Since f(-x) is not equal to –f(x), and it is not equal to f(x), it is neither even nor odd. Now, we will look at a vertical shift f(x) = (1/x) + b: ![]() First, we will look at a horizontal shift f(x) = 1/(x – a): However, a shifted reciprocal function is neither even nor odd. More generally, any function of the form f(x) = k/x is odd (for k not zero), since: The reciprocal function f(x) = 1/x has odd symmetry. Note the vertical asymptote at x = 1 and the horizontal asymptote at y = 3. It is the function f(x) = 1/x, but scaled by a factor of 2 (twice as tall), with a horizontal shift 1 unit to the right and a vertical shift of 3 units up. This is the graph of the shifted and scaled reciprocal function g(x) = (2/(x – 1)) + 3. You can see a graph of this scaled and shifted function below. For example, the function g(x) = (2/(x – 1)) + 3 looks like the reciprocal function f(x) = 1/x, but it is shifted 1 unit to the right, 3 units up, and stretched vertically by a factor of 2 (it is “twice as tall” as f(x) = 1/x). Of course, we can combine two or more of these shifts to get any variation we like. ![]() If b < 0, then the graph is shifted down by b units. If a 0, then the graph is shifted up by b units. If k 0, then the graph is shifted a units to the right. If k > 1, then the graph gets “taller” (stretched). f(x) = k/x scales the graph by a factor of k.We can take some variations of the basic reciprocal function f(x) = 1/x to scale or shift the graph as follows: In this case, y is the reciprocal of x, and we can also say that y is inversely proportional to x. Also note the vertical asymptote at x = 0 and the horizontal asymptote at y = 0. Note that two separate “parts” of the graph: top right and bottom left, which are disconnected. This is the graph of the reciprocal function, f(x) = 1/x or y = 1/x. You can see the graph of the reciprocal function f(x) = 1/x below: The “bottom left” part is to the left of the vertical asymptote x = 0 and below the horizontal asymptote y = 0.The “top right” part is to the right of the vertical asymptote x = 0 and above the horizontal asymptote y = 0.The reciprocal function f(x) = 1/x looks like it has two separate (disconnected) “parts”: What Does The Reciprocal Function Look Like? Remember that the reciprocal of any real number x (except x = 0) is 1/x. The function f(x) = 1/x is called the reciprocal function because it takes an input (x-variable) and gives the reciprocal 1/x as the output. The reciprocal function is also its own inverse function, since 1/(1/x) = x. ![]() The domain of the reciprocal function is the set of all real numbers, excluding zero. More generally, we say that y is inversely proportional to x if y = k/x for some real constant k (where k is not zero). This is a specific case of inverse variation between the variables y and x. We can also write the reciprocal function as The reciprocal function is a basic rational function with the form We’ll also look at examples of how to work backwards and find the equation of a reciprocal function from a graph or table. In this article, we’ll talk about reciprocal functions, what they are, and how to draw them. The same is true for any of its variations (scaled or shifted versions). Of course, the reciprocal function has both a horizontal and a vertical asymptote. The reciprocal function has variations, such as y = k/x (scaling or compression), y = 1/(x-k) (horizontal shift), and y = (1/x) + k (vertical shift). So, what is a reciprocal function? The reciprocal function has the form f(x) = 1/x or y = 1/x, and in this case, we can also say that y is inversely proportional to x. However, a reciprocal function can also describe a relationship between two variables where one increases as the other decreases. \), has an x-intercept, but no y-intercept unless it’s the line \(x=0\).The reciprocal of a number is used often in mathematics to solve algebra problems (with multiplicative inverses). ![]()
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