Limits/End Behavior
Determine the end behavior of the following function:
f(x) = x^6-7x^8+x^2+1
\lim_{x\to\infty} f(x) = \lim_{x\to\infty} -7x^8 = -\infty
\lim_{x\to-\infty} f(x) = \lim_{x\to-\infty} -7x^8 = -\infty
What is the domain of
f(x) = -e^x + 1?
\mathbb{R}
Find the inverse of the following function:
f(x) = \frac{3x+1}{5}
f^{-1}(x) = \frac{5x-1}{3}
Compute the following:
tan^{-1}(-1)=
-\frac{\pi}{4}
Is the following function one-to-one?
f(x) = cos(x) + 1
No
Compute the following limit:
\lim_{x\to \frac{\pi}{4}} \tan(x) +1=
2
What is the domain of the sequence generated by the following function:
a_n = \frac{1}{x+2}?
\mathbb{N}
Find a function that generates the following sequence:
\{-\frac{1}{3}, \frac{2}{9}, -\frac{3}{27}, \frac{4}{81},\ldots\}
a_n = \frac{(-1)^n n}{3^n}
Compute the following:
arcsin(sin(\frac{5\pi}{6}))=
\frac{\pi}{6}
Is the following rule a function: the rule that assigns each person to their birthday.
No, each day will have multiple people assigned to it (functions have one output per input).
Compute the following limit:
\lim_{x\to-\infty} -3(2^{-x})+1=
-\infty
What is the domain of
f(x) = \ln(2x+1) - \log_2(1-x)?
(-\frac{1}{2}, 1)
Solve the following equation for x:
\sqrt{2}\cos(x) +5 = 6
x = \frac{\pi}{4} + 2\pi k, k\in\mathbb{Z}
x = \frac{7\pi}{4} + 2\pi k, k\in \mathbb{Z}
Find the domain of the following function:
f(x) = \cos(\arcsin(-3x+2))
[\frac{1}{3}, 1]
Rewrite the following exponential equation in terms of logarithms:
3^{x+1} = 2y
\log_3(2y) = x+1
Determine all holes and asymptotes of the following rational function:
f(x) = \frac{x^2+4x+3}{x^2-2x-15}
HA at y=1, VA at x=5, and hole at (-3,\frac{1}{4})
What is the domain of
f(x) = \frac{x}{15x^2 - 7x -2}?
(-\infty, -\frac{1}{5}) \cup (-\frac{1}{5},\frac{2}{3})\cup(\frac{2}{3},\infty)
Solve the following equation for x:
\frac{3^{5x}}{3} = 9^{x+1}
x=1
What is the arc length swept out by an angle of \frac{\pi}{2} radians in a circle of radius 6?
3\pi
Which of the following best describes the function f(x) below: exponential, logarithmic, polynomial, or none of the above?
f(x) = \ln(5) + 3^{\log_3(x)}
Polynomial
List the limits you would need to compute to determine the end behavior of the following function:
f(x) = \log_5(x+2)-1
\lim_{x\to\infty} f(x)
\lim_{x\to -2^+} f(x)
What is the domain of
f(x) = \tan(x -\frac{\pi}{4})?
\{ x\in \mathbb{R} | x\ne \frac{3\pi}{4} + \pik, k\in \mathbb{Z}\}
Solve this equation for x:
\log_2(3x) +\log_2(x) = \log_2(12)
x=2
What is \cos\theta if \tan\theta = \frac{2}{3} and \pi < \theta < \frac{3\pi}{2} ?
\cos\theta = -\frac{3}{\sqrt{13}}
Which of the following terms describes the sequence listed below: alternating, strictly increasing, strictly decreasing, bounded from above, bounded from below, convergent, divergent?
\{-\frac{1}{3}, \frac{2}{9}, -\frac{3}{27}, \frac{4}{81},\ldots\}
Alternating, bounded from above, bounded from below, and convergent