Exam 4: Inverse, Exponential, and Logarithmic Functions

Write the word or phrase that best completes each statement or answers the question. -In the formula $\mathrm { A } ( \mathrm { t } ) = \mathrm { A } _ { 0 } \mathrm { e } ^ { \mathrm { kt } } , \mathrm { A }$ is the amount of radioactive material remaining from an initial amount $\mathrm { A } _ { 0 }$ at a given time $t$ , and $\mathrm { k }$ is a negative constant determined by the nature of the material A certain radioactive isotope decays at a rate of $0.225 \%$ annually. Determine half-life of this isotope, to the nearest year.

Match the function with its graph. - $f ( x ) = \log _ { 4 } 3 x$

Write an equivalent expression in exponential form. - $\log _ { \sqrt { 5 } } 625 = 8$

Graph the function. Give the domain and range. -

Determine whether the statement is true or false. -If a function f has an inverse function and the domain of f is all real numbers, then the domain of $\mathrm { f } ^ { - 1 }$ must also be all real numbers.

Write the word or phrase that best completes each statement or answers the question. -In the formula $\mathrm { N } = \mathrm { Ie } \mathrm { kt } , \mathrm { N }$ is the number of items in terms of an initial population I at a given time $\mathrm { t }$ and $\mathrm { k }$ is a growth constant equal to the percent of growth per unit time. There are currently 57 million cars in a certain country, decreasing by $2.2 \%$ annually. How many years will it take for this country to have 30 million cars? Round to the nearest year.

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

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

Write the word or phrase that best completes each statement or answers the question. -An artifact is discovered at a certain site. If it has $74 \%$ of the carbon-14 it originally contained, what is the approximate age of the artifact to the nearest year? (carbon-14 decays at the rate of $0.0125 \%$ annually.)

Find the value. Give an approximation to four decimal places. - $\log 513 - \log 379$

Determine whether the statement is true or false. -If f is a one-to-one function and the graph of f lies completely within the third quadrant, then the graph of $f ^{- 1}$ lies completely within the second quadrant.

Solve the equation. - $m ^ { - 4 } = \frac { 1 } { 256 }$

Determine whether or not the function is one-to-one. - $f ( x ) = ( 2 x - 7 ) ^ { 2 }$

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $\mathrm { f } ( \mathrm { x } ) = \left( \frac { 1 } { 4 } \right) ^ { \mathrm { x } }$ . Find $\mathrm { f } ( 3 )$ .

Solve the equation. Round to the nearest thousandth. - $\mathrm { e } ^ { 8 \mathrm { x } } \mathrm { e } ^ { 6 \mathrm { x } } = \mathrm { e } ^ { 2 }$

Solve the equation and express the solution in exact form. - $\log ( x - 9 ) = 1 - \log x$

Decide whether the pair of functions graphed are inverses. -

The graph of a function f is given. Use the graph to find the indicated value. - $f ^ { - 1 } ( 1 )$

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

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log b \sqrt { \frac { 4 x ^ { 3 } } { z ^ { 8 } } }$