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: Analyze Domain

The graph of the exponential function extends infinitely to the left and right along the x - axis. So, the domain (set of all x - values) is all real numbers. In inequality form, this is \(x\in(-\infty,\infty)\) or we can say "All reals" for the domain.

Step2: Analyze Range

The horizontal asymptote is a dashed line. From the graph, we can see that the function approaches the horizontal asymptote (let's assume the horizontal asymptote is \(y = 3\) from the graph's dashed line) from below and goes downwards? Wait, no, looking at the graph, the curve is increasing and approaches the dashed line (horizontal asymptote) from below. Wait, the y - values of the function: the horizontal asymptote is at \(y = 3\)? Wait, no, looking at the graph, the dashed line is at \(y = 3\)? Wait, the graph of the exponential function: the range is the set of all y - values. The horizontal asymptote is a dashed line, and the function's y - values are less than 3? Wait, no, looking at the graph, the curve starts from the bottom (low y - values) and approaches the dashed line (horizontal asymptote) as x increases. Wait, the y - intercept is at \(y = 2\)? Wait, maybe the horizontal asymptote is \(y=3\). Wait, the key is that for the range, we look at the possible y - values. The graph shows that the function never reaches or exceeds the horizontal asymptote (dashed line) and goes to negative infinity? Wait, no, the exponential function's graph: if it's an exponential growth or decay. Wait, the graph is increasing as x increases, and as x approaches \(-\infty\), y approaches \(-\infty\)? No, that can't be. Wait, maybe I misread. Wait, the graph: the x - axis is horizontal, y - axis vertical. The curve passes through ( - 4,0) maybe? Wait, no, the key is domain and range.

Wait, domain of an exponential function (in general, for functions like \(y = a^x + k\)): the domain is all real numbers because you can plug in any real number for x. So domain is all real numbers.

For the range: the horizontal asymptote is \(y = 3\) (from the dashed line). The function's y - values are less than 3? Wait, no, looking at the graph, the curve is above the x - axis? Wait, no, the curve goes down to the bottom left. Wait, maybe the horizontal asymptote is \(y = 3\), and the function's range is \(y<3\)? Wait, no, let's re - examine.

Wait, the problem says "the horizontal asymptote is shown as a dashed line". From the graph, the dashed line is at \(y = 3\) (since the grid has y - axis with marks, and the dashed line is at y = 3). The graph of the function is below the dashed line (y = 3) and extends to negative infinity? Wait, no, the curve is increasing. As x increases, y approaches 3 (the dashed line), and as x decreases (goes to the left), y decreases without bound (goes to \(-\infty\)). So the range is all real numbers less than 3? Wait, no, that would be \(y < 3\)? But exponential functions usually have ranges with a lower or upper bound. Wait, maybe the function is \(y=3 - e^{-x}\) or something, but regardless, from the graph:

Domain: All real numbers (because the graph extends infinitely left and right, so x can be any real number).

Range: The y - values are less than 3 (since the horizontal asymptote is y = 3 and the function approaches it from below, and as x decreases, y goes to \(-\infty\)). Wait, but maybe the horizontal asymptote is \(y = 3\), so the range is \(y<3\)? But let's confirm.

Wait, the domain of a function is the set of all x - values for which the function is defined. For exponential functions (in the form \(y = a^{x - h}+k\)), the domain is all real numbers because you can raise a to any real power. So domain is all real numbers.

For the range: if the horizontal asymptote is \(y = k\), then the range depends on the direction. If the function is \(y=a^{x}+k\), if \(a > 1\) (growth) or \(0 < a<1\) (decay). But from the graph, as x increases, y approaches the dashed line (y = 3), and as x decreases, y goes to \(-\infty\). So the range is \(y < 3\)? Wait, no, maybe the horizontal asymptote is \(y = 3\), and the function's range is \(y> - \infty\) and \(y < 3\), but that's equivalent to \(y\in(-\infty,3)\), so in inequality, \(y < 3\).

Wait, but let's check again. The domain: all real numbers (because the graph has no restrictions on x - values, it extends left and right infinitely). So domain is "All reals".

The range: the y - values are less than 3 (since the horizontal asymptote is at y = 3 and the function doesn't cross or go above it, and as x decreases, y goes to negative infinity). So range is \(y < 3\).

Answer:

(a) range: \(y < 3\)
(b) domain: All reals