Consider The Functions Below: F ( X ) = 4 X 2 + 11 X − 3 G ( X ) = 2 X + 1 \begin{array}{l} f(x)=4x^2+11x-3 \\ g(x)=2^x+1 \end{array} F ( X ) = 4 X 2 + 11 X − 3 G ( X ) = 2 X + 1 Which Statement Best Compares The Values Of The Two Functions?A. The Value Of G ( X G(x G ( X ] Always Exceeds The Value Of F ( X F(x F ( X ].B. The Value

Mar 13, 2025 by ADMIN 336 views

Comparing the Values of Two Functions: A Mathematical Analysis

In mathematics, functions are used to describe the relationship between variables. When comparing the values of two functions, it's essential to understand their behavior and characteristics. In this article, we will analyze two functions, $f(x)=4x^2+11x-3$ and $g(x)=2^x+1$ , and determine which statement best compares their values.

Function f(x)

The function $f(x)=4x^2+11x-3$ is a quadratic function, which means it has a parabolic shape. The coefficient of the squared term, 4, determines the direction and width of the parabola. In this case, the parabola opens upwards, indicating that the function has a minimum value.

To analyze the behavior of $f(x)$ , we can examine its derivative, $f'(x)=8x+11$ . The derivative represents the rate of change of the function, and its sign determines the direction of the function's slope. When $f'(x)>0$ , the function is increasing, and when $f'(x)<0$ , the function is decreasing.

Function g(x)

The function $g(x)=2^x+1$ is an exponential function, which means it has a rapid growth rate. The base of the exponent, 2, determines the rate of growth, and the exponent itself determines the direction of growth.

To analyze the behavior of $g(x)$ , we can examine its derivative, $g'(x)=2^x\ln(2)$ . The derivative represents the rate of change of the function, and its sign determines the direction of the function's slope. When $g'(x)>0$ , the function is increasing, and when $g'(x)<0$ , the function is decreasing.

Now that we have analyzed the behavior of both functions, let's compare their values. We are given two statements:

A. The value of $g(x)$ always exceeds the value of $f(x)$ . B. The value of $f(x)$ always exceeds the value of $g(x)$ .

To determine which statement is true, we need to examine the behavior of both functions over their entire domain.

Statement A

To prove that the value of $g(x)$ always exceeds the value of $f(x)$ , we need to show that $g(x)>f(x)$ for all $x$ in the domain of both functions.

Let's start by examining the behavior of $g(x)$ for large values of $x$ . As $x$ approaches infinity, $g(x)$ grows exponentially, and its value becomes much larger than $f(x)$ .

However, we also need to examine the behavior of $g(x)$ for small values of $x$ . As $x$ approaches negative infinity, $g(x)$ approaches 1, and its value becomes much smaller than $f(x)$ .

Since $g(x)$ has a rapid growth rate, it's possible that its value exceeds the value of $f(x)$ for some values of $x$ . However, we need to determine whether this is true for all $x$ in the domain of both functions.

Statement B

To prove that the value of $f(x)$ always exceeds the value of $g(x)$ , we need to show that $f(x)>g(x)$ for all $x$ in the domain of both functions.

Let's start by examining the behavior of $f(x)$ for large values of $x$ . As $x$ approaches infinity, $f(x)$ grows quadratically, and its value becomes much larger than $g(x)$ .

However, we also need to examine the behavior of $f(x)$ for small values of $x$ . As $x$ approaches negative infinity, $f(x)$ approaches negative infinity, and its value becomes much smaller than $g(x)$ .

Since $f(x)$ has a quadratic growth rate, it's possible that its value exceeds the value of $g(x)$ for some values of $x$ . However, we need to determine whether this is true for all $x$ in the domain of both functions.

In conclusion, we have analyzed the behavior of two functions, $f(x)=4x^2+11x-3$ and $g(x)=2^x+1$ , and compared their values. We have shown that the value of $g(x)$ always exceeds the value of $f(x)$ for large values of $x$ , but we have also shown that the value of $f(x)$ always exceeds the value of $g(x)$ for small values of $x$ .

Therefore, the correct statement is:

The value of $g(x)$ always exceeds the value of $f(x)$ for large values of $x$ , but the value of $f(x)$ always exceeds the value of $g(x)$ for small values of $x$ .

This statement is true because the exponential function $g(x)$ has a rapid growth rate, and its value becomes much larger than $f(x)$ for large values of $x$ . However, the quadratic function $f(x)$ has a slower growth rate, and its value becomes much smaller than $g(x)$ for small values of $x$ .

The final answer is:

A. The value of $g(x)$ always exceeds the value of $f(x)$ .

Note: This answer is only true for large values of $x$ . For small values of $x$ , the value of $f(x)$ always exceeds the value of $g(x)$ .
Q&A: Comparing the Values of Two Functions

In our previous article, we analyzed the behavior of two functions, $f(x)=4x^2+11x-3$ and $g(x)=2^x+1$ , and compared their values. We determined that the value of $g(x)$ always exceeds the value of $f(x)$ for large values of $x$ , but the value of $f(x)$ always exceeds the value of $g(x)$ for small values of $x$ .

In this article, we will answer some frequently asked questions about comparing the values of two functions.

Q: What is the main difference between the two functions?

A: The main difference between the two functions is their growth rate. The function $g(x)=2^x+1$ has an exponential growth rate, while the function $f(x)=4x^2+11x-3$ has a quadratic growth rate.

Q: Why does the value of $g(x)$ always exceed the value of $f(x)$ for large values of $x$ ?

A: The value of $g(x)$ always exceeds the value of $f(x)$ for large values of $x$ because the exponential function $g(x)$ has a rapid growth rate. As $x$ approaches infinity, $g(x)$ grows much faster than $f(x)$ .

Q: Why does the value of $f(x)$ always exceed the value of $g(x)$ for small values of $x$ ?

A: The value of $f(x)$ always exceeds the value of $g(x)$ for small values of $x$ because the quadratic function $f(x)$ has a slower growth rate. As $x$ approaches negative infinity, $f(x)$ approaches negative infinity, and its value becomes much smaller than $g(x)$ .

Q: Can we compare the values of two functions that have different growth rates?

A: Yes, we can compare the values of two functions that have different growth rates. However, we need to consider the behavior of both functions over their entire domain.

Q: How can we determine which function has a higher value for a given value of $x$ ?

A: To determine which function has a higher value for a given value of $x$ , we can substitute the value of $x$ into both functions and compare the results.

Q: Can we use calculus to compare the values of two functions?

A: Yes, we can use calculus to compare the values of two functions. We can examine the derivatives of both functions and compare their signs to determine which function is increasing or decreasing.

Q: What are some real-world applications of comparing the values of two functions?

A: There are many real-world applications of comparing the values of two functions. For example, in economics, we can compare the values of two functions to determine which one is more profitable. In engineering, we can compare the values of two functions to determine which one is more efficient.

In conclusion, comparing the values of two functions is an essential concept in mathematics. By understanding the behavior of both functions over their entire domain, we can determine which function has a higher value for a given value of $x$ . We can use calculus to compare the values of two functions and examine their derivatives to determine which function is increasing or decreasing. There are many real-world applications of comparing the values of two functions, and it is an essential tool for problem-solving in mathematics and other fields.

The final answer is:

The value of $g(x)$ always exceeds the value of $f(x)$ for large values of $x$ .
The value of $f(x)$ always exceeds the value of $g(x)$ for small values of $x$ .
We can compare the values of two functions that have different growth rates.
We can use calculus to compare the values of two functions.
There are many real-world applications of comparing the values of two functions.