- Explain why the graph of a rational function cannot cross its
vertical asymptote.
Answer: It cannot cross its vertical asymptote because the graph would be undefined at that value of x.
- Where does the rational function
cross its horizontal asymptote?
Answer: It crosses its horizontal asymptote y=3 at x=-8/5.
- If
estimate the value of the function when x=20,000,000,000,000,000 using its oblique (slant) asymptote. Will this estimate be more or less than the actual value of the function at x=20,000,000,000,000,000? Why?
Answer: Using the slant asymptote y=2x+1, we obtain the estimate
which will be less than the actual value of of the function because to get the actual value the remainder is subtracted from the asymptote.
.
- Find
.
Answer:
.
- State the range and domain of and g(x).
Answer: The domain of g(x) is x>4, or in interval notation,
.
The range of g(x) is all real numbers, or in interval notation,
.
The domain of
is all real numbers, or in interval
notation, .
The range of is y>4, or in interval notation,
.
- Sketch both functions on the same coordinate axis.
Answer:
- For which values of x is
?
Answer: They are equal only when x>0.
- For which values of x is
?
Answer: They are equal only when x>0.
- For which values of x is
?
Answer: They are equal only when x=1, or
.
- For which values of x is
?
Answer: They are equal only when x=1, or when
.
- Using the method of substitution,
Answer: x=0, and
.
- By sketching the graphs of both equations.
Answer:
Answer: It will take approximately 9 years and 10 months, or 9.8 years.