Select The Correct Answer From Each Drop-down Menu.Observe The Given Functions:$\[ \begin{align*} f(x) &= 4x + 3 \\ g(x) &= \left(\frac{5}{3}\right)^x \end{align*} \\]Complete The Sentences To Compare The Two Functions:1. Over The Interval

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Introduction

In mathematics, functions are used to describe the relationship between variables. Two functions, f(x) and g(x), are given, and we need to compare them over a specific interval. In this article, we will analyze the given functions and complete the sentences to compare them.

The Functions

The two functions given are:

  • f(x) = 4x + 3
  • g(x) = (5/3)^x

These functions are defined for all real numbers x.

Comparing the Functions

To compare the two functions, we need to analyze their behavior over the given interval. Let's start by understanding the characteristics of each function.

Linear Function: f(x) = 4x + 3

The function f(x) = 4x + 3 is a linear function. It has a constant rate of change, which is 4. This means that for every unit increase in x, the value of f(x) increases by 4 units.

Key Characteristics of f(x):

  • Domain: All real numbers x
  • Range: All real numbers y
  • Rate of Change: 4
  • Intercept: (0, 3)

Exponential Function: g(x) = (5/3)^x

The function g(x) = (5/3)^x is an exponential function. It has a base of 5/3, which is greater than 1. This means that the function will increase exponentially as x increases.

Key Characteristics of g(x):

  • Domain: All real numbers x
  • Range: All positive real numbers y
  • Rate of Change: Increases exponentially
  • Asymptote: y = 0 (horizontal asymptote)

Comparing the Functions

Now that we have analyzed the characteristics of each function, let's compare them over the given interval.

Interval: [0, ∞)

Over the interval [0, ∞), the function f(x) = 4x + 3 is increasing at a constant rate of 4. The function g(x) = (5/3)^x is increasing exponentially.

Comparison of f(x) and g(x) over [0, ∞):

  • f(x) < g(x) for all x in [0, ∞)
  • f(x) → ∞ as x → ∞
  • g(x) → ∞ as x → ∞

Interval: (-∞, 0)

Over the interval (-∞, 0), the function f(x) = 4x + 3 is decreasing at a constant rate of 4. The function g(x) = (5/3)^x is decreasing exponentially.

Comparison of f(x) and g(x) over (-∞, 0):

  • f(x) > g(x) for all x in (-∞, 0)
  • f(x) → -∞ as x → -∞
  • g(x) → 0 as x → -∞

Conclusion

In conclusion, the two functions f(x) = 4x + 3 and g(x) = (5/3)^x have different characteristics over the given interval. The linear function f(x) has a constant rate of change, while the exponential function g(x) has an increasing rate of change. Over the interval [0, ∞), f(x) < g(x), and over the interval (-∞, 0), f(x) > g(x).

Final Answer

Based on the analysis, the correct answer is:

  • f(x) < g(x) for all x in [0, ∞)
  • f(x) > g(x) for all x in (-∞, 0)

Introduction

In our previous article, we compared two functions, f(x) = 4x + 3 and g(x) = (5/3)^x, over a specific interval. In this article, we will answer some frequently asked questions related to the comparison of these two functions.

Q1: What is the difference between a linear function and an exponential function?

A1: A linear function has a constant rate of change, while an exponential function has an increasing or decreasing rate of change. In the case of f(x) = 4x + 3, the rate of change is constant at 4. In the case of g(x) = (5/3)^x, the rate of change is increasing exponentially.

Q2: How do the two functions behave over the interval [0, ∞)?

A2: Over the interval [0, ∞), the function f(x) = 4x + 3 is increasing at a constant rate of 4. The function g(x) = (5/3)^x is increasing exponentially. Therefore, f(x) < g(x) for all x in [0, ∞).

Q3: How do the two functions behave over the interval (-∞, 0)?

A3: Over the interval (-∞, 0), the function f(x) = 4x + 3 is decreasing at a constant rate of 4. The function g(x) = (5/3)^x is decreasing exponentially. Therefore, f(x) > g(x) for all x in (-∞, 0).

Q4: What is the relationship between the two functions over the entire real number line?

A4: The function f(x) = 4x + 3 is a linear function, while the function g(x) = (5/3)^x is an exponential function. Therefore, the relationship between the two functions over the entire real number line is that f(x) < g(x) for all x in (-∞, 0) and f(x) > g(x) for all x in [0, ∞).

Q5: Can we compare the two functions over other intervals?

A5: Yes, we can compare the two functions over other intervals. However, the relationship between the two functions may change depending on the interval. For example, if we consider the interval (-∞, -1), the function f(x) = 4x + 3 is decreasing at a constant rate of 4, while the function g(x) = (5/3)^x is increasing exponentially. Therefore, f(x) > g(x) for all x in (-∞, -1).

Q6: How do we determine the relationship between two functions?

A6: To determine the relationship between two functions, we need to analyze their characteristics, such as their rate of change, domain, and range. We also need to consider the interval over which we are comparing the functions.

Conclusion

In conclusion, the comparison of two functions involves analyzing their characteristics and determining their relationship over a specific interval. By understanding the characteristics of each function, we can determine the relationship between them and make conclusions about their behavior over different intervals.

Final Answer

Based on the Q&A, the correct answers are:

  • f(x) < g(x) for all x in [0, ∞)
  • f(x) > g(x) for all x in (-∞, 0)
  • The relationship between the two functions over the entire real number line is that f(x) < g(x) for all x in (-∞, 0) and f(x) > g(x) for all x in [0, ∞).