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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.

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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

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

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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.

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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.

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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?

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

y = 0.25 ( 3 ) x

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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
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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

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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?

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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.

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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?

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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)
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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.

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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.

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Questions & Answers

June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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