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Chapter 1: Problem 27

For each function, determine the largest possible domain. (a) \(f(x)=\frac{3}{x}+\frac{2}{3-x}-\frac{x}{2 x+2}\) (b) \(g(x)=\sqrt{x}+2 \sqrt{3-x}\)

Short Answer

Expert verified
The domain of function (a) \(f(x)=\frac{3}{x}+\frac{2}{3-x}-\frac{x}{2x+2}\) is all real numbers except -1, 0, and 3. The domain of function (b) \(g(x)=\sqrt{x}+2\sqrt{3-x}\) is all real numbers between and including 0 and 3.

Step by step solution

01

Analyzing the Function (a)

Let's start with function (a) \(f(x)=\frac{3}{x}+\frac{2}{3-x}-\frac{x}{2x+2}\). We need to find the domain, which are the values of x for which the function is defined. We cannot divide by zero, so we must exclude values of x that make the denominator zero in any of the terms. Hence, for each of the terms we get following conditions: \(x\neq0\), \(x\neq3\) and \(2x + 2\neq 0\) or \(x\neq -1\)
02

Conclusion for Function (a)

The largest possible domain for the function (a) is all real numbers except -1, 0, and 3.
03

Analyzing the Function (b)

Now, let's move to function (b) \(g(x)=\sqrt{x}+2\sqrt{3-x}\). Square root of a negative number is not a real number, so we can only take square root of a number if it's positive or zero. Therefore, the conditions for our domain become \(x\geq0\) and \(3-x\geq0\)
04

Conclusion for Function (b)

From the conditions, we find that \(x\geq0\) and also that \(x\leq3\). Therefore, the domain for function (b) is all real numbers between and including 0 and 3.

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Key Concepts

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

Calculus
When tackling calculus problems, understanding the domain of a function is crucial. Calculus is all about studying changes, and functions are often the objects undergoing these transformations. Considering the given problem, calculus aids us in determining the range of input values where the function can operate safely without causing undefined mathematical situations, like division by zero or taking the square root of a negative number.In more complex scenarios, calculus tools such as differentiation and integration would also require us to be mindful of the domain, as these processes are sensitive to the continuity and behavior of functions within their domains. In essence, the domain is the stage on which the calculus performance takes place, setting the boundaries for the show.
Function Analysis
Function analysis involves examining various aspects of a function to understand its characteristics and predict its behavior. The foundation of this analysis is identifying the domain, which tells us where the function 'lives'. In the exercise provided, we analyze two functions to identify their domains.By considering factors such as division by zero and the square root of negative numbers, we deduce the restrictions on the domain, thus determining the set of permissible 'x' values. This analysis is often the first step in the broader study of functions, which may involve examining their limits, continuity, differentiability, and integrability, all of which require a thorough understanding of the function's domain.
Real Numbers
The real numbers are the bread and butter of function domains in calculus and function analysis. They include all the numbers on the number line, encompassing both rational numbers (like fractions and integers) and irrational numbers (like the square root of two). When we say that the domain of a function is all real numbers except -1, 0, and 3, as in function (a) from the exercise, or that it is all real numbers between 0 and 3, inclusive, for function (b), we are specifically referring to those subsets of real numbers where the functions are well-defined and behave nicely.Ensuring that the values we select from the real numbers do not cause undefined behavior in functions is a key part of both setting up problems and finding solutions in calculus. In this way, the real numbers serve as the playing field for mathematical exploration and discovery.

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Most popular questions from this chapter

A gardener has a fixed length of fence that she will use to fence off a rectangular chili pepper garden. Express the area of the garden as a function of the length of one side of the garden. If you have trouble, reread the "Portable Strategies for Problem Solving" listed in this chapter. We've also included the following advice geared specifically toward this particular problem. Give the length of fencing a name, such as \(L\). (We don't know what \(L\) is, but we know that it is fixed, so \(L\) is a constant, not a variable.) \- Draw a picture of the garden. Call the length of one side of the fence \(s\). How can you express the length of the adjacent side in terms of \(L\) and \(s ?\) \- What expression gives the area enclosed by the fence?

Two bears, Bruno and Lollipop, discover a patch of huckleberries one morning. The patch covers an area of \(A\) acres and there are \(X\) bushels of huckleberries per acre. Bruno eats \(B\) bushels of huckleberries per hour; Lollipop can devour \(L\) bushels of huckleberries in \(C\) hours. Express your answers to parts (a) and (b) in terms of any or all of the constants \(A, X, B, L\), and \(C .\) (a) Express the number of bushels of huckleberries the two bears eat as a function of \(t\), the number of hours they have been eating. (b) In \(t\) hours, how many acres of huckleberries can the two bears together finish off? (c) Assuming that after \(T\) hours the bears have not yet finished the berry patch, how many hours longer does it take them to finish all the huckleberries in the patch? Express your answers in terms of any or all of the constants \(A, X, B, L, C\), and \(T\). If you are having difficulty, use this time-tested technique: Give the quantity you are looking for a name. (Avoid the letters already standing for something else.)

There are infinitely many prime numbers. This has been known for a long time; Euclid proved it sometime between 300 B.C. and 200 B.C. \({ }^{6}\) Number theorists (mathematicians who study the theory and properties of numbers) are interested in the distribution of prime numbers. Let \(P(n)=\) number of primes less than or equal to \(n\), where \(n\) is a positive number. Is \(P(n)\) a function? Explain.

If \(h(x)=\frac{x^{2}}{1-2 x}\), find (a) \(h(0)\) (b) \(h(3)\) (c) \(h(p+1)\) (d) \(h(3 p)\) (e) \(2 h(3 p)\) (f) \(\frac{1}{h(2 p)}\)

A typist can type \(W\) words per minute. On average, each computer illustration takes \(C\) minutes to create and \(I\) minutes to insert. (a) What is the estimated amount of time it will take for this typist to create a document \(N\) words long and containing \(Z\) illustrations? (b) The typist is paid \(\$ 13\) per hour for typing and a flat rate of \(\$ 10\) per picture. The cost of getting a document typed is a function of its length and the number of pictures. Write a function that gives a good estimate of the cost of getting a document of \(x\) words typed, assuming that the ratio of illustrations to words is \(1: 1000\). (c) Given the document described in part (b), express the typist's wages per hour as a function of \(x\).
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