Exam 12: Exponential Functions and Logarithmic Functions

Solve the problem. -The half-life of Antimony 111 is $2.9$ hours. If the formula $\mathrm { P } ( \mathrm { t } ) = \left( \frac { 1 } { 2 } \right) ^ { t / 2.9 }$ gives the percent (as a decimal) remaining after time $t$ (in hours), sketch P versus $t$ .

Find the inverse of the relation. -{(6, -6), (6, -5), (4, -4), (2, -3)}

Solve the problem. -An accountant tabulated a firm's profits for four recent years in the following table: The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimate future profits. Use the linear graph to estimate the profits in the year $2001 .$

Determine whether the function is one-to-one. - $f ( x ) = 7 x ^ { 2 } + 3$

Determine whether the function is one-to-one. - $f ( x ) = 4 x ^ { 2 } + x$

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line. - $y = 2 x ^ { 2 } + 4 x$

Determine whether the function is one-to-one. - $f ( x ) = \left( \frac { 1 } { 7 } \right) ^ { x }$

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line. - $y = 4 x ^ { 3 } + 7$

Determine whether the function is one-to-one. - $f ( x ) = x ^ { 3 } - 7$

Graph. - $f(x)=4^{3 x-2}$

Find f(x) and g(x) such that h(x) = $(f \circ g)(x)$ . - $h ( x ) = \frac { 8 } { \sqrt { 5 x + 4 } }$

Determine whether the function is one-to-one. - $f ( x ) = 16 - x ^ { 2 }$

Graph. - $x = 6 y$

Find f(x) and g(x) such that h(x) = $(f \circ g)(x)$ . - $h ( x ) = ( \sqrt { x } - 7 ) ^ { 2 }$

Solve the problem. -The half-life of a certain radioactive substance is 22 years. Suppose that at time $t = 0$ , there are $24 \mathrm {~g}$ of the substance. Then after t years, the number of grams of the substance remaining will be: $N ( t ) = 24 \left( \frac { 1 } { 2 } \right) ^ { t / 44 }$ How many grams of the substance will remain after 242 years? Round to the nearest hundredth when necessary.

Solve the problem. -Suppose that $90,000 is invested at 4% interest, compounded annually. Find a function A for the amount in the account after t years.

Solve the problem. -A size 10 dress in Country C is size 44 in Country D. A function that converts dress sizes in Country C to those in Country $D$ is $f ( x ) = 2 ( x + 12 )$ . Find a formula for the inverse of the function described.

Find the inverse of the relation. -{(-6, 2), (-2, 6), (9, -2), (-9, 2)}

Use composition to verify whether or not the inverse is correct. - $f ( x ) = 3 x + 6 , f ^ { - 1 } ( x ) = \frac { 1 } { 3 } x - 3$

Find an equation of the inverse of the relation. -y = 6x + 5