From the graph of the piecewise function, we have that:
a) The range of the function is (-∞,7).
b) The vertical asymptote is of x = 2, and the horizontal asymptotes are y = -1 and y = -2.
c) The end behavior of the function is described as follows: As x -> -∞, y -> -2 and as x -> ∞, y -> -1.
What is a piecewise function?
A piecewise function is a function that has different definitions, depending on the input.
For this problem, the definitions of the function are given as follows:
- [tex]f(x) = 3^{x+1} - 2, x \leq 1[/tex].
- [tex]f(x) = \frac{-x^2 + x + 2}{x^2 - 3x + 2}, x > 1[/tex]
The graph of the function is given at the end of the answer.
What is the range of the function?
The range of the function is the set that contains all the output values of the function, and in a graph, this is the values of y.
Hence:
The range of the function is (-∞,7).
What are the asymptotes of the function?
The asymptotes of the functions are the values of x for which the function is not defined. In this problem, the function is not defined for x = 2, hence the vertical asymptote is of x = 2.
The horizontal asymptotes are the values of the function when it goes to infinity, hence they are y = -2 and y = -1.
What is the end behavior of the function?
The end behavior is given by the limits of the function as it goes to infinity, being closely related to the horizontal asympotes.
Hence:
As x -> -∞, y -> -2 and as x -> ∞, y -> -1.
Please ignore the (60,1) point plotted on the graph, it should be (60,-1).
More can be learned about piecewise functions at https://brainly.com/question/27262465
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