Exam 5: Inverse, Exponential, and Logarithmic Functions

Use a graphing calculator to graph the function using the given viewing window. Use the graph to decide if the function is one-to-one. If the function is one-to-one, give the equation of the inverse function. - $f ( x ) = \frac { x + 1 } { x - 4 } ; [ - 10,10 ]$ by $[ - 10,10 ]$

Find the domain and range of the inverse of the given function. - $f ( x ) = \frac { 1 } { x + 4 }$

Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 2 x - 4 , \quad g ( x ) = \frac { x + 2 } { 4 }$

Let $\mathrm { u } = \ln \mathrm { a }$ and $\mathrm { v } = \ln \mathrm { b }$ . Write the following expression in terms of $\mathrm { u }$ and $\mathrm { v }$ without using the function $\ln$ . $\ln \sqrt [ 3 ] { \frac { a ^ { 9 } } { b ^ { 4 } } }$

Decide whether the pair of functions graphed are inverses. -

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

Solve the equation and express the solution in exact form. - $\log _ { 4 } \left( \log _ { 4 } x \right) = 1$

Evaluate the logarithm. - $\log _ { 6 } 6$

Evaluate the logarithm. - $\log _ { 2 } \frac { 1 } { 2 }$

A population is increasing according to the exponential function defined by $y = 3 \mathrm { e }$ - $04 \mathrm { x }$ , where $\mathrm { y }$ is in millions and $x$ is the number of years. Which of the following should be done in order to answer the question "When will the population reach 4 million?"

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

Suppose th $y = \frac { 2 - \log ( 100 - x ) } { 0.39 }$ can be used to calculate the number of years y for x percent of a population of 823 web-footed sparrows to die. Approximate the percentage (to the nearest whole per cent) of web-footed Sparrows that died after 2 yr.

An element decays at the rate of S(t) $S ( t ) = s e ^ { - 0.050 t }$ , where s is the initial amount in grams and t is the time in years since this initial amount was present. If you have a 13-gram piece of this element, how many grams will You have 3 years from now? Round your answer to the nearest tenth of a gram.

Solve the equation and express the solution in exact form. - $\ln ( 24 x - 8 ) = \ln 12$

Many states have passed laws against smoking in public places since January. The total number of states, N, that have passed a no smoking law, t months after January is given by the function $N ( t ) = \frac { 50 } { 1 + 22 e ^ { - 0.5 t } }$ How Many states had passed a law in January? Round to the nearest thousandth.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $f ( x ) = \left( \frac { 1 } { 5 } \right) ^ { x }$ . Find $f ( 3 )$ .

For the function as defined that is one-to-one, graph f and $\mathrm { f } ^ { - 1 }$ on the same axes. - $f ( x ) = \frac { 5 } { 2 } x + 2$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. - $4 ( 3 x - 1 ) = 18$

Let $\mathrm { u } = \ln \mathrm { a }$ and $\mathrm { v } = \ln \mathrm { b }$ . Write the following expression in terms of $\mathrm { u }$ and $\mathrm { v }$ without using the function ln. $\ln \left( b ^ { 2 } \sqrt [ 5 ] { a } \right)$

If the function is one-to-one, find its inverse. If not, write "not one-to-one." - $f ( x ) = x ^ { 3 } - 8$