Exam 5: Inverse, Exponential, and Logarithmic Functions

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 first and fourth quadrants, then the graph of $\mathrm { f } ^ { - 1 }$ lies completely within the first and second quadrants.

Decide whether the pair of functions graphed are inverses. -

Use the alphabet coding method $( A = 1 , B = 2 , C = 3 , \ldots Z = 26 )$ to solve the problem. -Encode the message BEWARE OF DOG using the one-to-one function defined by $f ( x ) = x ^ { 3 } - 1$ Give the inverse function that the decoder would need when the message is received.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = \left( \frac { 1 } { 6 } \right) ^ { x }$ . Find $f ( - 2 )$ .

Determine the function value. -Suppose $f ( x ) = \log _ { a } ( x )$ and $f ( 5 ) = 2$ . Find $f ( 125 )$ .

Write in logarithmic form. - $3 ^ { 2 } = 9$

Determine whether or not the function is one-to-one. - $f ( x ) = ( 4 x - 5 ) ^ { 2 }$

What is the rate on an investment that triples $9465 in 35 years? Assume interest is compounded monthly.

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

The growth in population of a city can be seen using the formula p(t) $\mathrm { p } ( \mathrm { t } ) = 3430 \mathrm { e } ^ { 0.007 \mathrm { t } }$ , where t is the number of years. Use this formula to calculate the population after 10 years.

Let $\mathrm { f } ( \mathrm { x } )$ compute the time in hours to travel $x$ miles at 49 miles per hour. What does $\mathrm { f } ^ { - 1 } ( \mathrm { x } )$ compute?

Find the function value. If the result is irrational, round your answer to the nearest thousandth. - $\text { Let } f ( x ) = 2 ^ { x } . \text { Find } f ( 3 )$

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $e ^ { x } + \ln x = 7$

The height in meters of women of a certain tribe is approximated by $h = 0.52 + 2 \log ( \mathrm { t } / 3 )$ where $t$ is the woman's age in years and $1 \leq t \leq 20$ . Estimate the height (to the nearest hundredth) of a woman of the tribe 4 years of age. (Round to the nearest hundredth.)

Graph the exponential function using transformations where appropriate. -

Solve the equation. - $\left( \frac { 1 } { 3 } \right) ^ { 5 x + 2 } = 9 ^ { x - 4 }$

If $\mathrm { f } ( \mathrm { x } ) = \mathrm { e } ^ { \mathrm { x } }$ , find $\mathrm { f } \left[ \ln \left( \frac { 1 } { \mathrm { e } ^ { 4 } } \right) \right]$

Solve the equation. - $( \sqrt { 2 } ) ^ { x + 4 } = 4 ^ { x }$

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.

Find the domain and range of the inverse of the given function. - $f ( x ) = - \frac { 1 } { x }$