Select The Correct Answer.Consider Functions { F $}$ And { G $} . . . { \begin{align} f(x) &= X^4 + 9x^2 - 3 \\ g(x) &= \left(\frac{1}{2}\right)^{x-2} \end{align} \} Using A Table Of Values, What Are The Approximate

Mar 10, 2025 by ADMIN 218 views

Selecting the Correct Answer: A Comprehensive Analysis of Functions f(x) and g(x)

In mathematics, functions are used to describe the relationship between variables. Two functions, f(x) and g(x), are given, and we are asked to find the approximate values of these functions using a table of values. In this article, we will delve into the world of functions and explore the properties of f(x) and g(x) to determine the correct answer.

Understanding Functions f(x) and g(x)

Function f(x)

The function f(x) is defined as:

f(x) = x^4 + 9x^2 - 3

This is a polynomial function of degree 4, which means it has a term with a degree of 4. The coefficients of the terms are 1, 9, and -3, respectively.

Function g(x)

The function g(x) is defined as:

g(x) = \left(\frac{1}{2}\right)^{x-2}

This is an exponential function with a base of 1/2 and an exponent of x-2.

Analyzing the Functions

To analyze the functions, we need to understand their behavior and how they change as x varies. Let's start by examining the function f(x).

Behavior of f(x)

The function f(x) is a polynomial function, which means it is continuous and differentiable everywhere. As x increases, the value of f(x) also increases. However, the rate of increase is not constant, and the function has a maximum value at x = 0.

Behavior of g(x)

The function g(x) is an exponential function, which means it is continuous and differentiable everywhere. As x increases, the value of g(x) decreases. The function has a minimum value at x = 2.

Creating a Table of Values

To find the approximate values of f(x) and g(x), we need to create a table of values. We will choose a range of x values and calculate the corresponding values of f(x) and g(x).

x	f(x)	g(x)
-2	-3	2
-1	-3	1.5
0	-3	1
1	7	0.5
2	13	0.25
3	23	0.125
4	37	0.0625

Analyzing the Table of Values

From the table of values, we can see that the function f(x) increases as x increases, while the function g(x) decreases as x increases. The function f(x) has a maximum value at x = 0, while the function g(x) has a minimum value at x = 2.

In conclusion, we have analyzed the functions f(x) and g(x) and created a table of values to find the approximate values of these functions. We have seen that the function f(x) is a polynomial function that increases as x increases, while the function g(x) is an exponential function that decreases as x increases. The function f(x) has a maximum value at x = 0, while the function g(x) has a minimum value at x = 2.

Based on the analysis of the functions and the table of values, we can conclude that the approximate values of f(x) and g(x) are:

f(x) ≈ 37 when x ≈ 4
g(x) ≈ 0.0625 when x ≈ 4

Therefore, the correct answer is:

f(x) ≈ 37 when x ≈ 4
g(x) ≈ 0.0625 when x ≈ 4
Selecting the Correct Answer: A Comprehensive Analysis of Functions f(x) and g(x)

Q: What is the difference between a polynomial function and an exponential function?

A: A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a coefficient and a variable raised to a non-negative integer power. An exponential function, on the other hand, is a function that can be written in the form of a^x, where a is a positive constant and x is the variable.

Q: What is the behavior of the function f(x) as x increases?

A: The function f(x) is a polynomial function of degree 4, which means it has a term with a degree of 4. As x increases, the value of f(x) also increases. However, the rate of increase is not constant, and the function has a maximum value at x = 0.

Q: What is the behavior of the function g(x) as x increases?

A: The function g(x) is an exponential function with a base of 1/2 and an exponent of x-2. As x increases, the value of g(x) decreases. The function has a minimum value at x = 2.

Q: How do we create a table of values for the functions f(x) and g(x)?

A: To create a table of values, we need to choose a range of x values and calculate the corresponding values of f(x) and g(x). We can use a calculator or a computer program to perform the calculations.

Q: What is the significance of the table of values in analyzing the functions f(x) and g(x)?

A: The table of values provides a visual representation of the functions f(x) and g(x) and helps us to understand their behavior and properties. It also allows us to identify the maximum and minimum values of the functions.

Q: How do we determine the approximate values of f(x) and g(x) from the table of values?

A: We can determine the approximate values of f(x) and g(x) by looking at the values in the table of values. For example, we can see that the value of f(x) is approximately 37 when x is approximately 4, and the value of g(x) is approximately 0.0625 when x is approximately 4.

Q: What is the final answer to the problem?

A: Based on the analysis of the functions and the table of values, we can conclude that the approximate values of f(x) and g(x) are:

f(x) ≈ 37 when x ≈ 4
g(x) ≈ 0.0625 when x ≈ 4

Therefore, the correct answer is:

f(x) ≈ 37 when x ≈ 4
g(x) ≈ 0.0625 when x ≈ 4

Common Mistakes to Avoid

Not understanding the difference between polynomial and exponential functions
Not creating a table of values to analyze the functions
Not identifying the maximum and minimum values of the functions
Not using a calculator or computer program to perform the calculations

Based on the analysis of the functions and the table of values, we can conclude that the approximate values of f(x) and g(x) are:

f(x) ≈ 37 when x ≈ 4
g(x) ≈ 0.0625 when x ≈ 4

Therefore, the correct answer is:

f(x) ≈ 37 when x ≈ 4
g(x) ≈ 0.0625 when x ≈ 4