Question

Describe the graph of the basic rational function, f(x)=1/x. Talk about its domain and range and compare the function to other ones, i.e. linear or quadratic or exponential.

Answer 1

F(x) = 1/x is a special function. is called the multiplicative inverse or the reciprocal function. the specific shape of the graph is called a rectangular hyperbola.

negative values of x with produce negative values of f(x) and positive values will produce positive function values. so this hyperbola is symmetric with respect to the origin and all negative values are in Q3 and all positive values are in Q1.

as x gets farther and farther from 0 in either direction, you can see that 1/x will be an ever-increasingly smaller distance from the x-axis. indeed, the limit of this function is zero as x approaches either positive or negative infinity.

if x is between -1 and 1, then the value of f(x) increases until x = 0, the limit of the function as x approaches 0 being positive and negative infinity depending on which direction you are coming from.

the domain is all real numbers with the exception of 0

the range also happens to be all real numbers with the exception of zero. all that really means is that zero has no multiplicative inverse.

strictly speaking, this is an exponential function, as the function is f(x) = x^(-1).