Problem 59 - Contemporary Precalculus

Question

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 \(\quad f(1)=1=g(1)\)

Step-by-Step Solution

Answer

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 1: 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).

Step 2: 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: $$ f(x) = ax^3+bx^2+cx $$

Step 3: 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: $$ f(x) = x^3 $$

Step 4: 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.

Step 5: 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.

Step 6: 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: $$ g(x) = \arctan(\tan(1) x) $$

Step 7: Conclusion

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

Key Concepts

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.