<< Chapter < Page Chapter >> Page >

Use a graphing utility to find an exponential regression formula f ( x ) and a logarithmic regression formula g ( x ) for the points ( 1.5 , 1.5 ) and ( 8.5 ,  8 .5 ) . Round all numbers to 6 decimal places. Graph the points and both formulas along with the line y = x on the same axis. Make a conjecture about the relationship of the regression formulas.

Got questions? Get instant answers now!

Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.

First rewrite the exponential with base e : f ( x ) = 1.034341 e 0 .247800x . Then test to verify that f ( g ( x ) ) = x , taking rounding error into consideration:

g ( f ( x ) ) = 4.035510 ln ( 1.034341 e 0 .247800x ) 0.136259 = 4.03551 ( ln ( 1.034341 ) + ln ( e 0 .2478 x ) ) 0.136259 = 4.03551 ( ln ( 1.034341 ) + 0 .2478 x ) 0.136259 = 0.136257 + 0.999999 x 0.136259 = 0.000002 + 0.999999 x 0 + x = x

Got questions? Get instant answers now!

Find the inverse function f 1 ( x ) for the logistic function f ( x ) = c 1 + a e b x . Show all steps.

Got questions? Get instant answers now!

Use the result from the previous exercise to graph the logistic model P ( t ) = 20 1 + 4 e 0.5 t along with its inverse on the same axis. What are the intercepts and asymptotes of each function?

Graph of P(t)=20/(1+40.5e^(-0.5t)) and P(t)=(ln(4)-ln((20/t)-1)/0.5.

The graph of P ( t ) has a y -intercept at (0, 4) and horizontal asymptotes at y = 0 and y = 20. The graph of P 1 ( t ) has an x - intercept at (4, 0) and vertical asymptotes at x = 0 and x = 20.

Got questions? Get instant answers now!

Chapter review exercises

Exponential Functions

Determine whether the function y = 156 ( 0.825 ) t represents exponential growth, exponential decay, or neither. Explain

exponential decay; The growth factor, 0.825 , is between 0 and 1.

Got questions? Get instant answers now!

The population of a herd of deer is represented by the function A ( t ) = 205 ( 1.13 ) t , where t is given in years. To the nearest whole number, what will the herd population be after 6 years?

Got questions? Get instant answers now!

Find an exponential equation that passes through the points (2, 2 .25) and ( 5 , 60.75 ) .

y = 0.25 ( 3 ) x

Got questions? Get instant answers now!

Determine whether [link] could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

x 1 2 3 4
f(x) 3 0.9 0.27 0.081
Got questions? Get instant answers now!

A retirement account is opened with an initial deposit of $8,500 and earns 8.12 % interest compounded monthly. What will the account be worth in 20 years?

$ 42 , 888.18

Got questions? Get instant answers now!

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with 7.5 % APR, compounded daily, in order to reach her goal in 3 years?

Got questions? Get instant answers now!

Does the equation y = 2.294 e 0.654 t represent continuous growth, continuous decay, or neither? Explain.

continuous decay; the growth rate is negative.

Got questions? Get instant answers now!

Suppose an investment account is opened with an initial deposit of $10,500 earning 6.25 % interest, compounded continuously. How much will the account be worth after 25 years?

Got questions? Get instant answers now!

Graphs of Exponential Functions

Graph the function f ( x ) = 3.5 ( 2 ) x . State the domain and range and give the y -intercept.

domain: all real numbers; range: all real numbers strictly greater than zero; y -intercept: (0, 3.5);

Graph of f(x)=3.5(2^x)
Got questions? Get instant answers now!

Graph the function f ( x ) = 4 ( 1 8 ) x and its reflection about the y -axis on the same axes, and give the y -intercept.

Got questions? Get instant answers now!

The graph of f ( x ) = 6.5 x is reflected about the y -axis and stretched vertically by a factor of 7. What is the equation of the new function, g ( x ) ? State its y -intercept, domain, and range.

g ( x ) = 7 ( 6.5 ) x ; y -intercept: ( 0 ,  7 ) ; Domain: all real numbers; Range: all real numbers greater than 0.

Got questions? Get instant answers now!

Questions & Answers

how fast can i understand functions without much difficulty
Joe Reply
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
no. should be about 150 minutes.
It should be 158.5 minutes.
ok, thanks
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
yeah, it does. why do we attempt to gain all of them one side or the other?
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro; thanks for putting it out there like that, 😁
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
dont forget the cube in each variable ;)
of she solves that, well ... then she has a lot of computational force under her command ....
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?