the graph of an exponential function is shown in the figure below. the horizontal asymptote is shown as a…
the graph of an exponential function is shown in the figure below. the horizontal asymptote is shown as a dashed line. find the domain and the range. 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.
Answer
Explanation:
Step1: Recall domain definition
The domain of a function is the set of all possible input - values (x - values). For an exponential function, there are no restrictions on the value of x.
Step2: Determine the domain
The domain is all real numbers. In inequality form, $x\in(-\infty,\infty)$ or $-\infty < x<\infty$.
Step3: Recall range definition
The range of a function is the set of all possible output - values (y - values). Looking at the graph, the function has a horizontal asymptote. The function values are always greater than the y - value of the horizontal asymptote.
Step4: Determine the range
Let the horizontal asymptote be at $y = k$. From the graph, the function values $y$ are greater than the y - value of the horizontal asymptote. If the horizontal asymptote is at $y = - 2$ (assuming from the general position in the graph), the range is $y>-2$.
Answer:
Domain: $-\infty < x<\infty$ Range: $y > - 2$ (assuming the horizontal asymptote is at $y=-2$)