Exam 12: Exponential Functions and Logarithmic Functions

Solve the problem. -The half-life of Cesium $134 \mathrm {~m}$ is $3.0$ hours. If the formula $\mathrm { P } ( \mathrm { t } ) = \left( \frac { 1 } { 2 } \right) ^ { \mathrm { t } / 3.0 }$ gives the percent (as a decimal) remaining after time $\mathrm { t }$ (in hours), sketch P versus $\mathrm { t }$ .

Determine whether the given function is one-to-one. If so, find a formula for the inverse. - $f ( x ) = x ^ { 3 } - 8$

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line. - $y = 2 x ^ { 3 } + 3$

Graph. - $x=\left(\frac{1}{4}\right)^{y}$

Find the requested composition of functions. -Given $f ( x ) = - 5 x + 2$ and $g ( x ) = 2 x + 4$ , find $g f ( x )$ .

Graph. - $f ( x ) = 3 ^ { 2 x - 1 }$

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

Determine whether the function is one-to-one. - $f ( x ) = 7 x ^ { 2 } - 6$

Solve the problem. -An accountant tabulated a firm's profits for four recent years in the following table: Year Profits 1996 \ 250,000 1997 \ 300,000 1998 \ 400,000 1999 \ 600,000 The accountant then fit both a linear graph and an exponential curve (seen below) to the data, in order to estimat profits. Use the exponential graph to estimate the profits in the year $2002 .$

Solve the problem. -The number of dislocated electric impulses per cubic inch in a transformer increases when lightning strikes by $\mathrm { D } ( \mathrm { x } ) = 1000 ( 2 ) ^ { \mathrm { x } }$ , where $\mathrm { x }$ is the time in milliseconds of the lightning strike. Find the number of dislocated impulses at $x = 0$ and $x = 5$ .

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

Determine whether the given function is one-to-one. If so, find a formula for the inverse. - $f ( x ) = 6 x ^ { 3 } + 4$

Determine whether the function is one-to-one. - $f ( x ) = x ^ { 2 } - 6$

Determine whether the function is one-to-one. - $f ( x ) = 64 - x ^ { 2 }$

Determine whether the function is one-to-one. - $f ( x ) = 6 x - 6$

Find f(x)and g(x)such that h(x)= (f ° g)(x). - $h ( x ) = | 7 x + 5 |$

Graph the equation of the relation using a solid line, and then graph the inverse of the relation using a dashed line. - $y = 7 x - 8$

Find an equation of the inverse of the relation. - $y = 2 x - 7$

Determine whether the function is one-to-one. - $f ( x ) = x ^ { 3 } + 8$

Graph the relation using solid circles and the inverse using open circles. - $\{ ( - 3 , - 12 ) , ( - 5 , - 11 ) , ( - 7 , - 10 ) , ( - 9 , - 9 ) \}$