The function is continuous and differentiable

Oct 13, 2025 by windailys.com 99 views

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will discuss a particular exponential function with the given conditions and determine the value of the base $b$ .

The Exponential Function

The exponential function is defined as $f(x) = b^x$ , where $b$ is the base and $x$ is the exponent. This function has several properties, including:

The base $b$ is a positive real number.
The function is continuous and differentiable.
The function has a horizontal asymptote at $y = 0$ .

Given Conditions

We are given that $f(0) = 1$ and $f(1) = 0.1$ . Using the definition of the exponential function, we can write:

$f(0) = b^0 = 1$
$f(1) = b^1 = 0.1$

Solving for the Base $b$

We can use the given conditions to solve for the base $b$ . From the first condition, we know that $b^0 = 1$ , which implies that $b = 1$ . However, this is not possible since $f(1) = 0.1$ , which is not equal to $1$ .

Let's try to solve for $b$ using the second condition. We have $b^1 = 0.1$ , which implies that $b = 0.1$ . However, this is not possible since $b$ must be a positive real number.

Alternative Approach

Let's try to solve for $b$ using an alternative approach. We can rewrite the exponential function as $f(x) = e^{x \ln b}$ , where $e$ is the base of the natural logarithm and $\ln b$ is the natural logarithm of $b$ . Using this form, we can write:

$f(0) = e^{0 \ln b} = 1$
$f(1) = e^{1 \ln b} = 0.1$

Solving for the Base $b$

We can use the above equations to solve for the base $b$ . From the first equation, we know that $e^{0 \ln b} = 1$ , which implies that $\ln b = 0$ . However, this is not possible since $b$ must be a positive real number.

Let's try to solve for $b$ using the second equation. We have $e^{1 \ln b} = 0.1$ , which implies that $\ln b = \ln 0.1$ . Taking the exponential of both sides, we get:

$b = e^{\ln 0.1} = 0.1$

Conclusion

In this article, we discussed a particular exponential function with the given conditions and determined the value of the base $b$ . We used two different approaches to solve for $b$ , and in both cases, we found that $b = 0.1$ .

Final Answer

The final answer is: $\boxed{0.1}$

Q&A

Q: What is the definition of an exponential function?

A: The exponential function is defined as $f(x) = b^x$ , where $b$ is the base and $x$ is the exponent.

Q: What are the properties of an exponential function?

A: The exponential function has several properties, including:

The base $b$ is a positive real number.
The function is continuous and differentiable.
The function has a horizontal asymptote at $y = 0$ .

Q: How do we solve for the base $b$ using the given conditions?

A: We can use the given conditions to solve for the base $b$ . From the first condition, we know that $b^0 = 1$ , which implies that $b = 1$ . However, this is not possible since $f(1) = 0.1$ , which is not equal to $1$ . Let's try to solve for $b$ using the second condition. We have $b^1 = 0.1$ , which implies that $b = 0.1$ . However, this is not possible since $b$ must be a positive real number.

Q: Can we use an alternative approach to solve for the base $b$ ?

A: Yes, we can use an alternative approach to solve for the base $b$ . We can rewrite the exponential function as $f(x) = e^{x \ln b}$ , where $e$ is the base of the natural logarithm and $\ln b$ is the natural logarithm of $b$ . Using this form, we can write:

$f(0) = e^{0 \ln b} = 1$
$f(1) = e^{1 \ln b} = 0.1$

Q: How do we solve for the base $b$ using the alternative approach?

A: We can use the above equations to solve for the base $b$ . From the first equation, we know that $e^{0 \ln b} = 1$ , which implies that $\ln b = 0$ . However, this is not possible since $b$ must be a positive real number. Let's try to solve for $b$ using the second equation. We have $e^{1 \ln b} = 0.1$ , which implies that $\ln b = \ln 0.1$ . Taking the exponential of both sides, we get:

$b = e^{\ln 0.1} = 0.1$

Q: What is the final answer?

A: The final answer is $\boxed{0.1}$ .

Q: Can you provide more examples of exponential functions?

A: Yes, here are a few examples of exponential functions:

$f(x) = 2^x$
$f(x) = 3^x$
$f(x) = 4^x$

Q: How do we graph an exponential function?

A: To graph an exponential function, we can use a graphing calculator or a computer program. We can also use a table of values to create a graph.

Q: What are some real-world applications of exponential functions?

A: Exponential functions have numerous real-world applications, including:

Population growth: Exponential functions can be used to model population growth.
Compound interest: Exponential functions can be used to calculate compound interest.
Radioactive decay: Exponential functions can be used to model radioactive decay.

Q: Can you provide more information about the properties of exponential functions?

A: Yes, here are some additional properties of exponential functions:

The function is one-to-one: This means that each value of $x$ corresponds to a unique value of $f(x)$ .
The function is monotonic: This means that the function is either increasing or decreasing.
The function has a horizontal asymptote at $y = 0$ : This means that as $x$ approaches infinity, $f(x)$ approaches 0.