The rational functions studied in this section all have the characteristic that the numerator and denominator do not share a common variable factor. We now investigate rational functions for which this is not the case. For Exercises 111–114, a. Write the domain of f in interval notation. b . Simplify the rational expression defining the function. c. Identify any vertical asymptotes. d . Identify any other values of x (other than those corresponding to vertical asymptotes) for which the function is discontinuous. e . Identify the graph of the function. f ( x ) = − x 2 + 2 x + 3 x + 1
The rational functions studied in this section all have the characteristic that the numerator and denominator do not share a common variable factor. We now investigate rational functions for which this is not the case. For Exercises 111–114, a. Write the domain of f in interval notation. b . Simplify the rational expression defining the function. c. Identify any vertical asymptotes. d . Identify any other values of x (other than those corresponding to vertical asymptotes) for which the function is discontinuous. e . Identify the graph of the function. f ( x ) = − x 2 + 2 x + 3 x + 1
Solution Summary: The author explains that the domain of the function f(x)=-x
The rational functions studied in this section all have the characteristic that the numerator and denominator do not share a common variable factor. We now investigate rational functions for which this is not the case. For Exercises 111–114,
a. Write the domain of f in interval notation.
b. Simplify the rational expression defining the function.
c. Identify any vertical asymptotes.
d. Identify any other values of x (other than those corresponding to vertical asymptotes) for which the function is discontinuous.
For Exercises 103–104, given y = f(x),
remainder
a. Divide the numerator by the denominator to write f(x) in the form f(x) = quotient +
divisor
b. Use transformations of y
1
to graph the function.
2x + 7
5х + 11
103. f(x)
104. f(x)
x + 3
x + 2
For Exercises 111–114, use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write θ as a function of x in three different ways. It is not necessary to rationalize the denominator.
Exercises 65–74: Use the graph of f to determine intervals
where f is increasing and where f is decreasing.
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