Increasing and Decreasing Exponential Functions

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Introduction

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In this article, we will explore the behavior of exponential functions, particularly those that decrease and approach negative infinity as x increases.

What are Exponential Functions?

An exponential function is a function that can be written in the form f(x) = ab^x, where a and b are constants, and x is the variable. The base b is a positive number, and it determines the rate at which the function grows or decays.

Increasing and Decreasing Exponential Functions

Exponential functions can be either increasing or decreasing, depending on the base b. If b > 1, the function is increasing, and if 0 < b < 1, the function is decreasing.

Decreasing Exponential Functions

A decreasing exponential function is a function that decreases as x increases. In other words, as the input x gets larger, the output of the function gets smaller.

Approaching Negative Infinity

A function that approaches negative infinity as x increases is a function that gets arbitrarily close to negative infinity as x gets larger. In other words, as x increases without bound, the function values get smaller and smaller, approaching negative infinity.

Selecting the Correct Answer

Now that we have a good understanding of exponential functions and their behavior, let's select the correct answer from the given options.

Option A: $f(x) = 3(6)^x - 2$

This function is an increasing exponential function, since the base 6 is greater than 1. As x increases, the function values will also increase, and they will not approach negative infinity.

Option B: $f(x) = 3(0.6)^x - 1$

This function is a decreasing exponential function, since the base 0.6 is less than 1. However, as x increases, the function values will approach negative infinity, but the function is not decreasing and approaching negative infinity.

Option C: $f(x) = -3(0.6)^x + 1$

This function is also a decreasing exponential function, since the base 0.6 is less than 1. However, the function is not decreasing and approaching negative infinity, but rather increasing and approaching positive infinity.

Option D: $f(x) = -3(6)^x + 2$

This function is a decreasing exponential function, since the base 6 is greater than 1, but the negative sign in front of the function makes it decrease. As x increases, the function values will approach negative infinity.

Conclusion

In conclusion, the correct answer is Option D: $f(x) = -3(6)^x + 2$ . This function is a decreasing exponential function that approaches negative infinity as x increases.

Key Takeaways

Exponential functions can be either increasing or decreasing, depending on the base b.
A decreasing exponential function is a function that decreases as x increases.
A function that approaches negative infinity as x increases is a function that gets arbitrarily close to negative infinity as x gets larger.
The correct answer is Option D: $f(x) = -3(6)^x + 2$ .

References

Larson, R., & Edwards, B. (2019). Calculus. Cengage Learning.
Stewart, J. (2019). Calculus: Early Transcendentals. Cengage Learning.
Anton, H. (2019). Calculus: A New Horizon. John Wiley & Sons.
Exponential Functions Q&A ==========================

Introduction

Exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In this article, we will answer some frequently asked questions about exponential functions.

Q: What is an exponential function?

A: An exponential function is a function that can be written in the form f(x) = ab^x, where a and b are constants, and x is the variable. The base b is a positive number, and it determines the rate at which the function grows or decays.

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

A: An exponential function is a function that can be written in the form f(x) = ab^x, where a and b are constants, and x is the variable. A polynomial function, on the other hand, is a function that can be written in the form f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, a_0 are constants, and n is a positive integer.

Q: What is the base of an exponential function?

A: The base of an exponential function is the constant b in the equation f(x) = ab^x. The base determines the rate at which the function grows or decays.

Q: What is the exponent of an exponential function?

A: The exponent of an exponential function is the variable x in the equation f(x) = ab^x. The exponent determines the input value of the function.

Q: What is the coefficient of an exponential function?

A: The coefficient of an exponential function is the constant a in the equation f(x) = ab^x. The coefficient determines the vertical shift of the function.

Q: What is the domain of an exponential function?

A: The domain of an exponential function is all real numbers, unless the base b is negative, in which case the domain is all real numbers except for the value that makes the exponent negative.

Q: What is the range of an exponential function?

A: The range of an exponential function is all positive real numbers, unless the base b is negative, in which case the range is all negative real numbers.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: What is the inverse of an exponential function?

A: The inverse of an exponential function is a logarithmic function. For example, if f(x) = 2^x, then the inverse function is f^(-1)(x) = log_2(x).

Q: What is the derivative of an exponential function?

A: The derivative of an exponential function is another exponential function. For example, if f(x) = 2^x, then the derivative is f'(x) = 2^x log(2).

Q: What is the integral of an exponential function?

A: The integral of an exponential function is another exponential function. For example, if f(x) = 2^x, then the integral is F(x) = (2^x)/log(2) + C.

Conclusion

In conclusion, exponential functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. We hope that this Q&A article has helped you to understand exponential functions better.

Key Takeaways

Exponential functions are a fundamental concept in mathematics.
Exponential functions can be written in the form f(x) = ab^x, where a and b are constants, and x is the variable.
The base b determines the rate at which the function grows or decays.
The exponent x determines the input value of the function.
The coefficient a determines the vertical shift of the function.
The domain of an exponential function is all real numbers, unless the base b is negative.
The range of an exponential function is all positive real numbers, unless the base b is negative.

References

Larson, R., & Edwards, B. (2019). Calculus. Cengage Learning.
Stewart, J. (2019). Calculus: Early Transcendentals. Cengage Learning.
Anton, H. (2019). Calculus: A New Horizon. John Wiley & Sons.