Question

the graph of an exponential function is shown in the figure below. the horizontal asymptote is shown as a dashed line. find the range and the domain. write your answers as inequalities, using x or y as appropriate. or, you may instead click on \empty set\ or \all reals\ as the answer. (a) range: (b) domain:

Explanation:

Step1: Determine the Domain

The domain of a function is the set of all possible x - values. For an exponential function, there are no restrictions on the input values (x - values) as we can plug in any real number for x. So, we check the graph: the graph extends infinitely to the left and right along the x - axis, meaning x can be any real number. So the domain is all real numbers.

Step2: Determine the Range

The range of a function is the set of all possible y - values. Looking at the graph, the horizontal asymptote is at \(y = 1\) (from the dashed line). The graph of the exponential function is above this horizontal asymptote and increases without bound as x increases. So, the y - values are all real numbers greater than 1. So we can write the range as \(y>1\) (or in terms of inequality with y, \(1 < y\) and y can go to infinity).

Answer:

(a) range: \(y > 1\)
(b) domain: All reals