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Chapter 3: Problem 30

The functions \(R(x), K(x), D(x)\), and \(L(x)\) are de ned as follows: $$R(x)=\frac{1}{x^{2}}, \quad K(x)=|x|, \quad D(x)=x+3, \quad L(x)=-5 x .$$ Evaluate the following expressions. (Be sure to give simpli ed expressions whenever possible.) (a) \(R(K(L(x)))\) (b) \(R(L(R(x)))\) (c) \(R(K(x))\) (d) \(R(D(R(x)))\)

Short Answer

Expert verified
(a) \(1/25x^{2}\); (b) \(1/25x^{4}\); (c) \(1/x^{2}\); (d) \(1/(1/x^{2} + 3)^{2}\).

Step by step solution

01

- Evaluating \(R(K(L(x)))\)

We begin by evaluating \(L(x) = -5x\), producing \(-5x\) as the result. Next, we calculate \(K(x)\), where \(x = -5x\), giving us \(|-5x|\). The absolute value bars would turn any negative result positive. Therefore, the answer is \(5x\). Lastly, we put this result into \(R(x)\), yielding \(R(5x)\). As per the definition, \(R(x) = 1/x^{2}\). Replace \(x\) with \(5x\) results in \(1/(5x)^{2}\) or \(1/25x^{2}\).
02

- Evaluating \(R(L(R(x)))\)

Initially, we run \(R(x)\), which produces \(1/x^{2}\) as the result. Then, we input the result into \(L(x)\), obtaining \(-5*1/x^{2}\), or \(-5/x^{2}\). Finally, plug into \(R(x)\) to get \(1/(-5/x^{2})^{2}\), which simplifies to \(1/25x^{4}\).
03

- Evaluating \(R(K(x))\)

At first, we find \(K(x) = |x|\). Therefore, plug this into \(R(x)\) to get \(1/|x|^{2}\). As any number squared will be positive, the absolute value bars are redundant, and we get \(1/x^{2}\).
04

- Evaluating \(R(D(R(x)))\)

We start off with \(R(x)\) which is \(1/x^{2}\). Then we use this as input to \(D(x)\) to have \(1/x^{2} + 3\). Afterwards, input this into \(R(x)\) resulting in \(1/(1/x^{2} + 3)^{2}\).

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Key Concepts

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

Algebraic Functions
Algebraic functions are essential components in the study of mathematics and involve operations such as addition, subtraction, multiplication, division, and taking roots. Among the functions we've been evaluating, each algebraic function follows a specific formula or rule that determines its output for given inputs. For example:
  • In the original exercise, we examine several algebraic functions with different rules, such as multiplying by a constant, taking the absolute value, or transforming with fractions.
  • Each of these functions, like \(L(x) = -5x\) or \(D(x) = x + 3\), fits the classification of algebraic functions due to their reliance on basic arithmetic and algebraic operations.
Understanding the behavior of algebraic functions is pivotal as they create the foundation for more complex mathematical concepts such as rational, exponential, and logarithmic functions. Algebraic functions help us model and simplify real-world problems by using mathematical expressions.
Absolute Value Function
The absolute value function, often represented as \(|x|\), is crucial in transforming any number into its non-negative counterpart. The primary role of this function is to measure the distance from zero, ensuring the result is always positive. The exercise introduces us to this idea:
  • Taking \(K(x) = |x|\) for instance, whenever we substitute \(-5x\) from \(L(x)\), the absolute value function makes \(-5x\) become \(5x\).
  • This transformation is vital because, in many contexts, both the magnitude and sign of a number matter, while in others, only the magnitude is significant.
The absolute value function is also graphically represented as a 'V' shape, which reflects its symmetrical property around the y-axis. This makes it particularly useful in problems where distance and magnitude are important.
Rational Functions
Rational functions are expressed as the ratio of two polynomial functions, represented by the general form \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\). In our exercise, \(R(x) = \frac{1}{x^2}\) is a simple example of a rational function:
  • This function takes any given input \(x\) and transforms it by squaring \(x\) and taking the reciprocal.
  • Rational functions like this can be used to describe inversely proportional relationships or distribute resources efficiently in various scenarios.
The complexity of rational functions can increase with the degree of the polynomials involved, making function evaluation a significant skill. Evaluating compositions such as \(R(K(L(x)))\) demonstrates how outputs of a rational function can drastically change with different compositions, and why understanding their behavior is essential.
Function Evaluation
Function evaluation involves calculating the output of a function given a specific input. It is a fundamental process in algebra and higher mathematics:
  • In the exercise, learners are tasked with evaluating various compositions like \(R(K(L(x)))\), \(R(L(R(x)))\), and others by methodically applying each function to the result of the previous one.
  • This involves understanding how to correctly substitute inputs and simplify expressions in order to find the final, simplified result.
The practice of function evaluation is crucial not only for solving mathematical problems but also for learning to analyze and interpret real-world phenomena using mathematical models. Through these evaluations, students understand relationships between variables and improve their computational skills, preparing themselves for more advanced topics.

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Most popular questions from this chapter

The Cambridge Widget Company is producing widgets. The xed costs for the company (costs for rent, equipment, etc.) are $$\$ 20,000.$$ This means that before any widgets are produced, the company must spend $$\$ 20,000.$$ Suppose that each widget produced costs the company an additional $$\$ 10 .$$ Let \(x\) equal the number of widgets the company produced. (a) Write a total cost function, \(C(x)\), that gives the cost of producing \(x\) widgets. (Check that your function works, e.g., check that \(C(1)=20,010\) and \(C(2)=20,020 .)\) Graph \(C(x)\). (b) At what rate is the total cost increasing with the production of each widget? In other words, nd \(\Delta C / \Delta x\). (c) Suppose the company sells widgets for $$\$ 50$$ each. Write a revenue function, \(R(x)\), that tells us the revenue received from selling \(x\) widgets. Graph \(R(x)\). (d) Pro \(\mathrm{t}=\) total revenue \(-\) total cost, so the pro \(\mathrm{t}\) function, \(P(x)\), which tells us the pro \(\mathrm{t}\) the company gets by producing and selling \(x\) widgets, can be found by computing \(R(x)-C(x) .\) Write the pro \(\mathrm{t}\) function and graph it. (e) Find \(P(400)\) and \(P(700)\); interpret your answers. Find \(P(401)\) and \(P(402) .\) By how much does the pro t increase for each additional widget sold? Is \(\Delta P / \Delta x\) constant for all values of \(x\) ? (f) How many widgets must the company sell in order to break even? (Breaking even means that the pro \(\mathrm{t}\) is \(0 ;\) the total cost is equal to the total revenue.) (g) Suppose the Cambridge Widget Company has the equipment to produce at maximum 1200 widgets. Then the domain of the pro \(\mathrm{t}\) function is all integers \(x\) where \(0 \leq x \leq 1200\). What is the range? How many widgets should be produced and sold in order to maximize the company s pro ts?

Let \(f(x)=x^{2}\) and \(g(x)=1 / x\). Use your knowledge of the graphs of \(f\) and \(g\) to sketch the graph of \(h(x)=f(x) \cdot g(x)\). Where is \(h(x)\) unde ned? How can you indicate this on your graph? How does your graphing calculator deal with the point at which \(h\) is unde ned?

Let \(f(x)=|x|, g(x)=\sqrt{x}\), and \(h(x)=x-2\). Find the domain for each of the following. (a) \(p(x)=h(g(h(x)))\) (b) \(q(x)=f(h(g(x)))\)

Decompose the functions by finding functions \(f(x)\) and \(g(x)\), \(f(x) \neq x\) and \(g(x) \neq x\), such that \(h(x)=f(g(x))\). $$ h(x)=(\sqrt{x})^{3}-2 \sqrt{x}+3 $$

Find \(h(x)=f(g(x))\) and \(j(x)=g(f(x)) .\) What are the domains of \(h\) and \(j\) ? $$ f(x)=\frac{x}{x-3} \text { and } g(x)=\frac{2}{x} $$
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