MATH 107 University of Maryland College Algebra Problem Exercises

I’ve attached the document required in regards to the problems I need assistance with. The subjects in question are Radicals and Equations, and Functions.

Template 5
1. The surface area s of a highly expandable, advertisement beach ball (a sphere) is a
function of its diameter d and is given by the formula s(d) = π d2, where π  3.14.
Suppose that the last season’s beach ball, with some filling gas remaining in it, is to be
inflated for the new season and its diameter increases with time due to inflation
according to the formula d(t) =
is measured in inches.
𝟏 2
t
𝟑
+ 10, where t is measured in seconds, t ≥ 0, and d
Showing your work, answer the following questions (round final answers to two decimal
places and provide units):
(a) What is the diameter of the beach ball before inflation begins for the new season?
(Do not forget the unit.)
(b) What is the surface area of the beach ball before inflation begins for the new
season? (Do not forget the unit.)
(c) What is the diameter of the beach ball for an inflation time of nine seconds? (Do not
forget the unit.)
(d) What is the surface area of the beach ball for an inflation time of nine seconds? (Do
not forget the unit.)
(e) Find and interpret (s ○ d)(t).
(f) Using your answer to Part (e), and showing your work, find (s ○ d)(0) as a factor of π
first, and then in decimal format with unit.
(g) Using your answer to Part (e), and showing your work, find (s ○ d)(9) as a factor of π
first, and then in decimal format with unit.
(h) Supporting your answer briefly, show the mathematical domain of the function (s ○
d)(t) in interval notations.
(i) Showing your work, determine the physical/applied (real life) domain of the function
(s ○ d)(t) within the context of the problem, knowing that, according to the
manufacturer, the beach ball will explode when its diameter exceeds 85 inches?
3.
(a) Write the function h(x) = 3×4 − 1 as a composition of non-identity functions q(x) and
r(x). In other words, find r(x) and q(x) such that (r○q)(x) would result in the h(x), not
choosing any identity function.
(b) Check your answer by composing your q(x) with r(x).
4. (a) How, based on the graph of a function, we can tell that the function has an inverse?
(b) Apply your prescription to the following graphs and state which one, if any, is
invertible.
(c) In case an invertible function is noted, draw the graph of its inverse.
5. (a) Without resorting to graphing, and before applying the inverse-finding steps, how can
we determine whether the function given below has an inverse? Show your work.
𝑔(𝑥) =
𝑥
𝑥−7
(b) Find the inverse function 𝒈−𝟏 (x).
(c) To check your answer to Part (b), it is required to show
(𝑔○ 𝑔−1 )(𝑥) = 𝑥 and (𝑔−1 ○ g)(x) = x
(d) Write the domain and the range of g(x) in interval notations.
(e) Write the domain and the range of 𝑔−1 (𝑥) in interval notations.
6. Find exact solution(s) for the following equations:
2 + √3𝑥 + 10 = −𝑥
7.
(a)
Suppose that you are given the graph of a base b exponential function, drawn on the x-y
plane. Briefly explain how the value of base b can be found from the graph.
(b)
Given the equation 𝟑−𝒙
(c)
Given the equation ln(x) = 100, find x.
(d)
A calculator cannot directly find log12 (235) but can give you the base-10 logarithm and/or
the natural logarithm of 235. Showing your work, compute the value of log12 (235) using
the capability of that calculator.
= 𝟒, carefully convert it to a logarithmic equation and find x.
(e)
Using the figures and Theorem on page 423 of the e-book, set up a sign chart for the log
function with a base greater than 1 and a sign chart for the log function with a base
between 0 and 1, each reflecting the signs, the zero of the function, and the asymptote.
(f)
For a value of x > 1, do we have log(x) = ln(x), log(x) < ln(x), or log(x) > ln(x)? Support your
answer briefly, without using any numerical value for x.