Exam 4: 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 third and fourth quadrants, then the graph of f-1 lies completely within the second and third quadrants.

Find the function value. If the result is irrational, round your answer to the nearest thousandth. -Let $g ( x ) = \left( \frac { 1 } { 3 } \right) ^ { x }$ . Find $g ( 4.5 )$ .

Write the word or phrase that best completes each statement or answers the question. -Suppose $f ( x ) = 32.1 + 1.2 \log ( x + 1 )$ models salinity of ocean water to depths of 1000 meters at a certain latitude. $x$ is the depth in meters and $f ( x )$ is in grams of salt per kilogram of seawater. Approximate the depth (to the nearest tenth of a meter) where the salinity equals 34 .

Write the word or phrase that best completes each statement or answers the question. -A population is increasing according to the exponential function defined by $\mathrm { y } = 3 \mathrm { e } ^{.04 x }$ , 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 "How large will the population be in 4 years?"

Write the expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers. - $\log _ { 2 } 13 - \log _ { 2 } m$

Solve the problem. -An earthquake was recorded with an intensity which was $1,258,925$ times more powerful than a reference level earthquake, or $1,258,925 \cdot \mathrm { I } 0$ . What is the magnitude of this earthquake on the Richter scale (rounded to the nearest tenth)? Intensity on the Richter scale is $\log _ { 10 } ( \mathrm { I } / \mathrm { I } 0 )$ .

Solve the problem. -Let $\mathrm { f } ( \mathrm { x } )$ compute the cost of a rental car after $x$ days of use at $\$ 41$ per day. What is the interpretation of the solution of $f ^ { - 1 } ( x ) = 147$ ?

Decide whether or not the functions are inverses of each other.Use the definition of inverses to determine whether f and g are inverses. - $f ( x ) = 9 x ^ { 2 } + 9 , x \geq 0 ; \quad g ( x ) = \sqrt { \frac { x - 9 } { 9 } } , x \geq 9$

Evaluate the logarithm. - $\log _ { 5 } \frac { 1 } { 25 }$

Use the change of base rule to find the logarithm to four decimal places. - $\log _ { 3 } 0.739$

What are the domain and range for the equation y = 2x?

Solve the equation. - $x = \log _ { 10 } 0.01$

Solve the problem. -Suppose $f ( x ) = 28.4 + 1.3 \log ( x + 1 )$ models salinity of ocean water to depths of 1000 meters at a certain latitude. $x$ is the depth in meters and $f ( x )$ is in grams of salt per kilogram of seawater. Approximate the salinity (to the nearest hundredth) when the depth is 383 meters.

Solve the following graphically. If necessary, round answers to the nearest thousandth. - $500 ( 1 + 0.02 ) ^ { 12 x } = 750$

Solve for the indicated variable. - $\mathrm { P } = 20,000 \cdot \mathrm { e } ^ { \mathrm { t } / 3 }$ , for $\mathrm { t }$

Solve the problem. -Given that $f ( x ) = e ^ { x - 1 } + 3$ , find $f ^ { - 1 } ( x )$ and give the domain and range of $f ^ { - 1 } ( x )$ .

Use the properties of logarithms to rewrite the expression. Simplify the result if possible. Assume all variables represent positive real numbers. - $\log _ { 8 } \left( \frac { \sqrt { 7 } } { 9 } \right)$

Use a graphing calculator to estimate the solution set of the equation. Round to the nearest hundredth. -2(2x - 1) = 14

Determine whether or not the function is one-to-one. -

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