SOLUTION: This is the last of my quiz questions that I am using to prep for my final exam. For the function defined by f(x)=5x-4, find a formula for f-^1(x). Solve the equation 4^(2x)

For the function defined by f(x)=5x-4, find a formula for f-^1(x). 


Replace  by 

Interchange x and y

Solve for y



Replace  by  


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Solve the equation 4^(2x+1) = 2^(3x+6) ?

 

Write  as 



Simplify the left side





Now use the principle that if two powers of the same 
positive base are equal, and the base is not 1, then
the exponents must be equal:





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Find an exponential function of the form  with
y-intercept 2, horizontal asymptote y=-2, that passes through the 
point P(1,4).

The graph of  has horizontal asymptote y = 0, the 
x-axis. To move that asymptote down to y=-2 we must shift the graph 
downward by 2 units, so we add -2 to the right side, which makes 
c = -2

So far we have  

To have y-intercept 2, it must pass through the point (0,2)
so that f(0) = 2.






So far we have 

To pass through the point (1,4) we have f(1) = 4.
 





So we have 

 

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Solve the equation (f o g)^-1(3)?

You didn't give f(x) and g(x), so I can't do that one
without that information.

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For the function defined by , , use 
a sketch to help find a formula for .

First we sketch the graph of , 

  

Now we draw in the identity line whose equation is 



Now we draw the curve reflecting it into the identity line and we 
get the

  

The blue curve is the graph of 

To find an equation for  we proceed as in the
first problem above


Replace  by 

Interchange x and y

Solve for y

=±
We can see from the graph that the blue curve is above the
x-axis so we ignore the - solution
Replace  by  


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The amount of a radioactive tracer remaining after t days is given by
A = A0e-0.058t, where A0 is the 
starting amount at the beginning of the time period. How many days 
will it take for one half of the original amount to decay?

This is when A = A0/2

So substitute A0/2 for A

A0/2 = A0e-0.058t 

Divide both sides by A0

1/2 = e-0.058t

Use principle that X = eY can be written as Y = ln(X)
to rewrite that as

-0.058t = ln(1/2)
      t = ln(1/2)/(-0.058) = 11.95081346 days or about 12 days.

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The population P of a certain culture is expected to be given by 
a model P=100ert where r is a constant to be determined 
and t is a number of days since the original population of 100 was 
established. Find the value of r if the population is expected to 
reach 200 in 3 days.

Substitute 200 for P and 3 for t

P = 100ert
200 = 100er·3
200 = 100e3r
Divide both sides by 100
2 = e3r
Use principle that X = eY can be written as Y = ln(X)
to rewrite that as
3r = ln(2)
 r = ln(2)/3 = .2310490602

Edwin