MATH 107 UMGC Inequalities Algebraic Functions & Function Dominion Exam Practice

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I need detailed solutionsfor all the questions in the attached file.

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Name: ____________________
Date: _____________________
MATH 107
Quiz 5
3
1) For f (x) = √x + 1, g(x) = 4×2 −x find:
a. (g ◦ f )(0)
b. (f ◦ g)( 21 )
c. (f ◦ f )(− 2)
2) Use the given pair of functions f (x) = 3 − x2 , g(x) = √x + 1
to find and simplify the expression (g ◦ f )(x) and state the
domain of each using interval notation.
3) Show that the given function is one-to-one and find its
inverse. Check your answers algebraically and graphically.
f (x) =
x
1 − 3x
.
4) The price of a media player is given as a function of the
weekly sales x according to the formula p(x) = 550 − 30x for
0 ≤ x ≤ 40.
a. Find p−1 (x) and state its domain.
b. Find and interpret p−1 (125) .
c. Find x if p−1 (x) = 0
5) Perform the indicated operations and simplify.
√x + x−1
√x
6) Find all real solutions for 2x − 1 = √x + 3 .
7) Solve the inequality 12 − √x − 3 ≤ 15 .
8) Evaluate the expressions.
a. ln(e4 )
1
b. log 6 ( 36
)
c. log 13 (√13)
d. ln(426log(1) )
9) For f (x) = ex , g(x) = 10 − e−x , sketch the graph of y = g(x)
by starting with the graph of y = f(x) and using
transformations. Track at least three points of your choice
and the horizontal asymptote through the transformations.
State the domain and range of g(x).
10) Earthquakes are complicated events and it is not our
intent to provide a complete discussion of the science
involved in them. Instead, we refer the interested reader to a
solid course in Geology or the U.S. Geological Survey’s
Earthquake Hazards Program found here and present only a
simplified version of the Richter scale. The Richter scale
measures the magnitude of an earthquake by comparing the
amplitude of the seismic waves of the given earthquake to
those of a “magnitude 0 event”, which was chosen to be a
seismograph reading of 0.001 millimeters recorded on a
seismometer 100 kilometers from the earthquake’s
epicenter. Specifically, the magnitude of an earthquake is
x
given by M (x) = log ( 1000
) where x is the seismograph
reading in millimeters of the earthquake recorded 100
kilometers from the epicenter.
a. Show that M(0.001) = 0.
b. Compute M(70,000).
c. If the magnitude of the earthquake was 6.7 on the
Richter scale, what was the seismograph reading?