Exam 4: Inverse, Exponential, and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. -A certain country's Consumer Price Index is approximated by $\mathrm { A } ( \mathrm { t } ) = 100 \mathrm { e } ^ { 0.024 \mathrm { t } }$ , where $t$ represents the number of years. Use the function to determine the year in which costs will be $50 \%$ higher than in year 0 .

Write the word or phrase that best completes each statement or answers the question. -Suppose the consumption of electricity grows at $4.5 \%$ per year, compounded continuously. Find the number of years before the use of electricity has tripled. Round the answer to the nearest hundredth.

Determine whether the statement is true or false. -Any function of the form $f ( x ) = x ^ { n }$ where n is an odd integer, has an inverse.

Find the domain and range of the inverse of the given function. -f(x) = 8x + 6

If the function is one-to-one, find its inverse. If not, write ʺnot one-to-one.ʺ - $f ( x ) = 5 x ^ { 2 } - 9 , x \geq 0$

Solve the equation. - $5 x - 35 = \log _ { x } 1$

Write the word or phrase that best completes each statement or answers the question. -The number of books in a small library increases according to the function $\mathrm { B } = 2200 \mathrm { e } 0.03 \mathrm { t }$ , where is measured in years. How many books will the library have after 5 years?

Write the word or phrase that best completes each statement or answers the question. -Use the formula $\mathrm { P } = \mathrm { Ie } \mathrm { kt }$ . A bacterial culture has an initial population of 10,000 . If its population declines to 4000 in 2 hours, what will it be at the end of 4 hours?

Determine whether or not the function is one-to-one. - $f ( x ) = \sqrt [ 3 ] { 6 x - 5 }$

Write the word or phrase that best completes each statement or answers the question. -In year 0, the number of female athletes participating in Summer Olympic-Type Games was 500 . In year 11, about 3550 participated in the Summer Olympics in Atlanta. Assuming that $\mathrm { P } ( 0 ) = 500$ and that the exponential model applies, find the value of $\mathrm { k }$ rounded to the hundredths place, and write the function.

Write an equation for the graph given. The graph represents an exponential function f with base 2 or 3, translated and/or reflected. -

Solve for the indicated variable. - $\mathrm { V } - \mathrm { V } _ { 0 } = \left( \mathrm { V } _ { 1 } - \mathrm { V } _ { 0 } \right) 10 ^ { - \mathrm { kt } }$ , for $\mathrm { t }$

Solve the problem. -The population of country A in millions is modeled by $\mathrm { f } ( \mathrm { x } ) = 17.8 \mathrm { e } 0.0018 \mathrm { x }$ . During the same time period, the population of country $B$ in millions is modeled by $g ( x ) = 13.7 e ^ { 0.0136 x }$ . In both formulas $x$ is the number of years. Assuming these trends continue, estimate what the population will be when the populations are equal. Round to the nearest million.

Find the domain and range of the inverse of the given function. - $f ( x ) = - \frac { 7 } { x }$

Write an equation for the graph given. The graph represents an logarithmic function f with base 2 or 3, translated and/or reflected. -

Solve the equation and express the solution in exact form. - $\log ( 3 + x ) - \log ( x - 4 ) = \log 4$

Write the word or phrase that best completes each statement or answers the question. -The graph of $\mathrm { y } = 10 ^ { \mathrm { x } }$ is shown with the coordinates of a point displayed at the bottom of the screen. Write the logarithmic equation associated with the display.

Evaluate the logarithm. - $\log _ { 3 } \frac { 1 } { 3 }$

Solve the problem. -The decibel level $\mathrm { D }$ of a sound is related to its intensity I by $\mathrm { D } = 10 \log \left( \frac { \mathrm { I } } { \mathrm { I } _ { 0 } } \right)$ . If $\mathrm { I } _ { 0 }$ is $10 ^ { - 12 }$ , then what is the intensity of a noise measured at 41 decibels? Express your answer in scientific notation, rounding to three significant digits, if necessary.

Solve the equation. - $\log _ { 7 } \sqrt { 7^6 } = x$