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Chapter 3: Problem 59

Give an example of two different functions \(f\) and \(g\) that have all of the following properties: $$ f(-1)=1=g(-1) \quad \text { and } \quad f(0)=0=g(0) $$ and \(\quad f(1)=1=g(1)\)

Short Answer

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Question: Find two different functions, f and g, such that f(-1) = 1, f(0) = 0, f(1)=1, g(-1) = 1, g(0) = 0, and g(1) = 1. Answer: Two functions that satisfy these conditions are f(x) = x^3 and g(x) = arctan(tan(1) x).

Step by step solution

01

Choose a function type for f

Let's choose a polynomial function for f. An odd degree polynomial ensures that the function f would pass through the three given points (-1, 1), (0, 0), and (1, 1).
02

Define the function f

Let's set the basic form for f to be a cubic (degree 3) polynomial. The most straightforward choice is to use a basic cubic function with no additional constant: $$ f(x) = ax^3+bx^2+cx $$
03

Find the coefficients for f

We need to find the coefficients a, b, and c such that the given conditions hold: 1. f(-1) = 1: \(a(-1)^3+b(-1)^2+c(-1) = -a+b-c=1\) 2. f(0) = 0: \(0a+0b+0c = 0\) 3. f(1) = 1: \(a(1)^3+b(1)^2+c(1) = a+b+c=1\) Solving the above system of equations, we see that a good choice is to have a = 1, b = 0, and c = 0. This makes our function f(x) as: $$ f(x) = x^3 $$
04

Choose a function type for g

In order to make g different from f, let's choose a trigonometric function for g. An odd function that can pass through the three given points is tangent. But to make it satisfy the given conditions, let's consider the arctangent function.
05

Define the function g

We can define the function g as: $$ g(x) = \arctan(kx) $$ where k is a constant that will make g fulfill the given conditions.
06

Find the value for k

In order to satisfy the given conditions, we need to find the value of k: 1. g(-1) = 1: \(\arctan(k(-1)) = 1\) 2. g(0) = 0: \(\arctan(k(0)) = 0\) 3. g(1) = 1: \(\arctan(k(1)) = 1\) From the second condition, we can see that k can take any real value since the arctan of 0 is always 0. If we take the arctan of both sides of the first condition, we get k as: $$ k = \tan(1) $$ The same will hold for the third condition. Thus, our function g can be defined as: $$ g(x) = \arctan(\tan(1) x) $$
07

Conclusion

We have found two different functions, f and g, that satisfy the given properties: $$ f(x) = x^3 \quad \text{and} \quad g(x) = \arctan(\tan(1) x) $$

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Key Concepts

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

Polynomial Functions
Polynomial functions are a fundamental concept in mathematics. They are expressions consisting of variables and coefficients. A polynomial can have different degrees, represented by the highest power of the variable. Here are a few important points about polynomial functions:
  • A polynomial function can be represented as: \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \). The terms \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients.
  • The degree of the polynomial is the largest exponent, which determines its basic shape and behavior.
  • Polynomial functions are continuous and smooth, meaning they don't have any breaks or sharp turns.
To achieve the conditions given in the original problem, we used a cubic polynomial function, which is a specific type of polynomial.
Trigonometric Functions
Trigonometric functions relate angles of a triangle to the ratios of its sides. They are periodic, meaning they repeat their values in regular intervals.
Trigonometric functions are essential in modeling situations involving cycles or waves, such as sound and light waves. The key trigonometric functions include:
  • Sine and Cosine: These functions are used to determine the coordinates of points on a unit circle.
  • Tangent: Represents sine divided by cosine. It's undefined where cosine is zero.
  • Arctangent: It's the inverse of the tangent function, often denoted as \( \arctan(x) \).
For the function \( g(x) \) in the problem, we utilized the arctangent. This choice makes it possible to satisfy the given conditions for a different function that still aligns with the constraints.
Cubic Functions
Cubic functions belong to the family of polynomial functions and are defined by a third-degree expression.
The general form of a cubic function is \( f(x) = ax^3 + bx^2 + cx + d \). Here are features of a cubic function:
  • They can have up to three real roots, due to the degree being three.
  • Cubic functions can have an inflection point, where the concavity of the curve changes.
  • The number and nature of the roots can vary depending on the coefficients.
In the example problem, the function \( f(x) = x^3 \) was chosen. A simplified cubic function with no degree 2 or degree 1 terms was used to meet the specified values and conditions.

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Most popular questions from this chapter

Determine the domain of the function according to the usual convention. $$k(u)=\sqrt{u}$$

The estimated number of 15 - to 24 -year-old people worldwide (in millions) who are living with HIV/AIDS in selected years is given in the table."$$\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & 2001 & 2003 & 2005 & 2007 & 2009 \\ \hline \begin{array}{l} \text { 15- to 24-year-olds } \\ \text { with HIV/AIDS } \end{array} & 12 & 14.5 & 17 & 19 & 20.5 \\ \hline \end{array}$$(a) Use linear regression to find a function that models this data, with \(x=0\) corresponding to 2000 (b) According to your function, what is the average rate of change in this HIV/AIDS population over any time period between 2001 and \(2009 ?\) (c) Use the data in the table to find the average rate of change in this HIV/AIDS population from 2001 to 2009 . How does this rate compare with the one given by the model? (d) If the model remains accurate, when will the number of people in this age group with HIV/AIDS reach 25 million?

Compute and simplify the difference quotient of the function. $$f(x)=x^{2}+3 x-1$$

Find the dimensions of a box with a square base that has a volume of 867 cubic inches and the smallest possible surface area, as follows. (a) Write an equation for the surface area \(S\) of the box in terms of \(x\) and \(h .[\) Be sure to include all four sides, the top, and the bottom of the box.] (b) Write an equation in \(x\) and \(h\) that expresses the fact that the volume of the box is 867 . (c) Write an equation that expresses \(S\) as a function of \(x\). [Hint: Solve the equation in part (b) for \(h\), and substitute the result in the equation of part (a).] (d) Graph the function in part (c), and find the value of \(x\) that produces the smallest possible value of \(S .\) What is \(h\) in this case?

Use algebra to find the inverse of the given one-to-one function. $$f(x)=1 / x$$
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