/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Let \(g\) be the function define... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 8

Let \(g\) be the function defined by \(g(u)=(3 u-2)^{3 / 2}\). Find \(g(1), g(6), g\left(\frac{11}{3}\right)\), and \(g(u+1)\)

Short Answer

Expert verified
g(1) = 1, g(6) = 64, g(11/3) = 27, g(u+1) = \((3u + 1)^{\frac{3}{2}}\)

Step by step solution

01

Evaluate g(1)

To find the value of g(1), we can plug in 1 for u in the given equation \(g(u)=(3u-2)^{\frac{3}{2}}\): \[g(1) = (3(1) - 2)^{\frac{3}{2}}\] \[g(1) = (3 - 2)^{\frac{3}{2}}\] \[g(1) = (1)^{\frac{3}{2}} = 1\]
02

Evaluate g(6)

Now let's find the value of g(6) by plugging in 6 for u: \[g(6)=(3(6)-2)^{\frac{3}{2}}\] \[g(6)=(18-2)^{\frac{3}{2}}\] \[g(6)=(16)^{\frac{3}{2}}\] \[g(6)= (2^4)^{\frac{3}{2}} = 2^{\frac{4\times3}{2}} = 2^6 = 64\]
03

Evaluate g(11/3)

Next, let's find the value of \(g\left(\frac{11}{3}\right)\) by plugging in \(\frac{11}{3}\) for u: \[g\left(\frac{11}{3}\right) = \left(3\left(\frac{11}{3}\right) - 2\right)^{\frac{3}{2}}\] \[g\left(\frac{11}{3}\right) = (11 - 2)^{\frac{3}{2}}\] \[g\left(\frac{11}{3}\right) = (9)^{\frac{3}{2}}\] \[g\left(\frac{11}{3}\right) = (3^2)^{\frac{3}{2}} = 3^3 = 27\]
04

Find g(u+1)

Lastly, we need to find an expression for the value of g(u+1). To do this, we can replace u with (u+1) in the given equation and then simplify: \(g(u+1) = (3(u + 1) - 2)^{\frac{3}{2}}\) \(g(u+1) = (3u + 3 - 2)^{\frac{3}{2}}\) \[g(u+1) = (3u + 1)^{\frac{3}{2}}\] So, the expressions we found are: g(1) = 1 g(6) = 64 g(11/3) = 27 g(u+1) = \((3u + 1)^{\frac{3}{2}}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mathematical Functions
A mathematical function is a relationship between a set of inputs and a set of possible outputs where each input is related to exactly one output. You can think of it like a machine that has an input and an output: for every input value, the function gives you an output value. In the exercise, the function is represented as g(u) where u is the input variable, and the output is determined by the algebraic expression (3u - 2)^{3/2}.

Evaluating a function means finding the result of the function for a given value of the input variable. When the exercise asks for g(1), g(6), or g(11/3), it's looking for the outputs when the input is 1, 6, and 11/3 respectively. Similarly, to find g(u+1), you're asked to determine the function's output when the input is u+1 instead of just u. By substituting the input values into the function and simplifying, we obtain the corresponding outputs.
Exponents
An exponent is a number that indicates how many times to multiply a base number by itself. The operation is known as exponentiation. In the expression (3u - 2)^{3/2}, the exponent 3/2 signifies that after calculating the base (3u - 2), it must be raised to the power of 1.5 (since 3/2 is 1.5). In simpler terms, that base is first squared, and then the square root is taken (since squaring is the exponent 2, and the square root corresponds to the exponent 1/2).

Understanding Fractional Exponents

Fractional exponents, like the one in the exercise, might seem confusing at first glance. A fractional exponent like a^{b/c} means the base a is raised to the power of b, followed by taking the c-th root of the result. The procedure used in the exercise simplifies the calculation by breaking down the exponent into more manageable parts, especially when dealing with integers and their known powers.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables (like u), and arithmetic operations (like addition, subtraction, multiplication, and division). In the context of the exercise, (3u - 2)^{3/2} is an algebraic expression where u is the variable, and it's involved in multiplication by 3, subtraction of 2, and exponentiation. These operations are done following the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Manipulating Algebraic Expressions

To solve algebraic expressions, you must be comfortable with various arithmetic rules and properties, like the distributive property used to expand g(u+1) in the fourth step of the solution. When evaluating functions or simplifying expressions, it鈥檚 essential to perform the operations correctly according to PEMDAS and to be familiar with how to manipulate exponents effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 1-18, find the vertex, the \(x\) -intercepts (if any), and sketch the parabola. \(f(x)=x^{2}+x-6\)

DECISION ANALYSIS A product may be made using machine I or machine II. The manufacturer estimates that the monthly fixed costs of using machine I are \(\$ 18,000\), whereas the monthly fixed costs of using machine II are \(\$ 15,000\). The variable costs of manufacturing 1 unit of the product using machine I and machine II are \(\$ 15\) and \(\$ 20\), respectively. The product sells for \(\$ 50\) each. a. Find the cost functions associated with using each machine. b. Sketch the graphs of the cost functions of part (a) and the revenue functions on the same set of axes. c. Which machine should management choose in order to maximize their profit if the projected sales are 450 units? 550 units? 650 units? d. What is the profit for each case in part (c)?

For the supply equations in Exercises 27 and 28, where \(x\) is the quantity supplied in units of a thousand and \(p\) is the unit price in dollars, (a) sketch the supply curve and (b) determine the price at which the supplier will make 2000 units of the commodity available in the market. \(p=2 x^{2}+18\)

MARKET EQuILIBRIUM The weekly demand and supply functions for Sportsman \(5 \times 7\) tents are given by $$ \begin{array}{l} p=-0.1 x^{2}-x+40 \\ p=0.1 x^{2}+2 x+20 \end{array} $$ respectively, where \(p\) is measured in dollars and \(x\) is measured in units of a hundred. Find the equilibrium quantity and price.

BREAK-EvEN ANALYsIS A division of Carter Enterprises produces "Personal Income Tax" diaries. Each diary sells for \(\$ 8\). The monthly fixed costs incurred by the division are \(\$ 25,000\), and the variable cost of producing each diary is \(\$ 3\). a. Find the break-even point for the division. b. What should be the level of sales in order for the division to realize a \(15 \%\) profit over the cost of making the
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.