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[T] The concentration of hydrogen ions in a substance is denoted by $\left[{\text{H}}^{+}\right],$ measured in moles per liter. The pH of a substance is defined by the logarithmic function $\text{pH}=\text{\u2212}\text{log}\left[{\text{H}}^{+}\right].$ This function is used to measure the acidity of a substance. The pH of water is 7. A substance with a pH less than 7 is an acid, whereas one that has a pH of more than 7 is a base.
i. a. pH = 8 b. Base ii. a. pH = 3 b. Acid iii. a. pH = 4 b. Acid
[T] Iodine-131 is a radioactive substance that decays according to the function $Q\left(t\right)={Q}_{0}\xb7{e}^{\mathrm{-0.08664}t},$ where ${Q}_{0}$ is the initial quantity of a sample of the substance and t is in days. Determine how long it takes (to the nearest day) for 95% of a quantity to decay.
[T] According to the World Bank, at the end of 2013 ( $t=0$ ) the U.S. population was 316 million and was increasing according to the following model:
$P\left(t\right)=316{e}^{0.0074t},$
where P is measured in millions of people and t is measured in years after 2013.
a. $~333$ million b. 94 years from 2013, or in 2107
[T] The amount A accumulated after 1000 dollars is invested for t years at an interest rate of 4% is modeled by the function $A\left(t\right)=1000{(1.04)}^{t}.$
[T] A bacterial colony grown in a lab is known to double in number in 12 hours. Suppose, initially, there are 1000 bacteria present.
a. $k\approx 0.0578$ b. $\approx 92$ hours
[T] The rabbit population on a game reserve doubles every 6 months. Suppose there were 120 rabbits initially.
[T] The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time, in Japan, an earthquake with magnitude 4.9 caused only minor damage. Approximately how much more energy was released by the San Francisco earthquake than by the Japanese earthquake?
The San Francisco earthquake had ${10}^{3.4}\text{or}\phantom{\rule{0.2em}{0ex}}~2512$ times more energy than the Japan earthquake.
True or False ? Justify your answer with a proof or a counterexample.
A function is always one-to-one.
A relation that passes the horizontal and vertical line tests is a one-to-one function.
A relation passing the horizontal line test is a function.
False
For the following problems, state the domain and range of the given functions:
$f={x}^{2}+2x-3,\phantom{\rule{3em}{0ex}}g=\text{ln}(x-5),\phantom{\rule{3em}{0ex}}h=\frac{1}{x+4}$
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