Which Of These Is Another Way To Write The Function F ( X ) = 5 X F(x) = 5^x F ( X ) = 5 X ?A. F(x) = \frac{1}{125}\left(625^x\right ] B. F ( X ) = ( 625 X ) 1 4 F(x) = \left(625^x\right)^{\frac{1}{4}} F ( X ) = ( 62 5 X ) 4 1 ​ C. F ( X ) = ( 625 X ) 1 125 F(x) = \left(625^x\right)^{\frac{1}{125}} F ( X ) = ( 62 5 X ) 125 1 ​ D. $f(x)

Introduction

Exponential functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and economics. The function $f(x) = 5^x$ is a classic example of an exponential function, where the base is 5 and the exponent is $x$. In this article, we will explore alternative ways to represent this function, focusing on the options provided in the discussion category.

Understanding Exponential Functions

Before we dive into the alternatives, let's briefly review the concept of exponential functions. An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. The base $a$ determines the growth rate of the function, and the exponent $x$ determines the magnitude of the output.

Option A: $f(x) = \frac{1}{125}\left(625^x\right)$

Let's examine the first option: $f(x) = \frac{1}{125}\left(625^x\right)$. To determine if this is an equivalent representation of the function $f(x) = 5^x$, we need to analyze the components of the expression.

• The base of the expression is $625$, which is equal to $5^4$. This means that the base of the expression is actually $5^4$.
• The exponent of the expression is still $x$, which remains unchanged.
• The coefficient of the expression is $\frac{1}{125}$, which can be rewritten as $\frac{1}{5^3}$.

By substituting the base and coefficient into the expression, we get:

$$f(x) = \frac{1}{5^3}\left(5^4\right)^x$$

Using the properties of exponents, we can simplify the expression:

$$f(x) = \frac{1}{5^3}\cdot 5^{4x}$$

Applying the rule of exponents that states $a^m \cdot a^n = a^{m+n}$, we can combine the exponents:

$$f(x) = 5^{4x-3}$$

This expression is equivalent to the original function $f(x) = 5^x$, but with a different exponent. Therefore, option A is a valid alternative representation of the function.

Option B: $f(x) = \left(625^x\right)^{\frac{1}{4}}$

Let's examine the second option: $f(x) = \left(625^x\right)^{\frac{1}{4}}$. To determine if this is an equivalent representation of the function $f(x) = 5^x$, we need to analyze the components of the expression.

• The base of the expression is $625$, which is equal to $5^4$. This means that the base of the expression is actually $5^4$.
• The exponent of the expression is $x$, which remains unchanged.
• The coefficient of the expression is $\frac{1}{4}$, which can be rewritten as $\frac{1}{5^{\log_5 4}}$.

By substituting the base and coefficient into the expression, we get:

f(x) = \left(5^4\right)^x\right)^{\frac{1}{4}}

Using the properties of exponents, we can simplify the expression:

$$f(x) = \left(5^{4x}\right)^{\frac{1}{4}}$$

Applying the rule of exponents that states $\left(a^m\right)^n = a^{mn}$, we can combine the exponents:

$$f(x) = 5^{4x \cdot \frac{1}{4}}$$

Simplifying the exponent, we get:

$$f(x) = 5^x$$

This expression is equivalent to the original function $f(x) = 5^x$. Therefore, option B is a valid alternative representation of the function.

Option C: $f(x) = \left(625^x\right)^{\frac{1}{125}}$

Let's examine the third option: $f(x) = \left(625^x\right)^{\frac{1}{125}}$. To determine if this is an equivalent representation of the function $f(x) = 5^x$, we need to analyze the components of the expression.

• The base of the expression is $625$, which is equal to $5^4$. This means that the base of the expression is actually $5^4$.
• The exponent of the expression is $x$, which remains unchanged.
• The coefficient of the expression is $\frac{1}{125}$, which can be rewritten as $\frac{1}{5^3}$.

By substituting the base and coefficient into the expression, we get:

f(x) = \left(5^4\right)^x\right)^{\frac{1}{125}}

Using the properties of exponents, we can simplify the expression:

$$f(x) = \left(5^{4x}\right)^{\frac{1}{125}}$$

Applying the rule of exponents that states $\left(a^m\right)^n = a^{mn}$, we can combine the exponents:

$$f(x) = 5^{4x \cdot \frac{1}{125}}$$

Simplifying the exponent, we get:

$$f(x) = 5^{\frac{4x}{125}}$$

This expression is not equivalent to the original function $f(x) = 5^x$. Therefore, option C is not a valid alternative representation of the function.

Conclusion

In conclusion, we have explored three alternative representations of the function $f(x) = 5^x$. We found that options A and B are valid alternative representations of the function, while option C is not.

• Option A: $f(x) = \frac{1}{125}\left(625^x\right)$ is a valid alternative representation of the function, with a different exponent.
• Option B: $f(x) = \left(625^x\right)^{\frac{1}{4}}$ is a valid alternative representation of the function, with the same exponent.
• Option C: $f(x) = \left(625^x\right)^{\frac{1}{125}}$ is not a valid alternative representation of the function.

Q: What is an exponential function?

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. The base $a$ determines the growth rate of the function, and the exponent $x$ determines the magnitude of the output.

Q: What are some common examples of exponential functions?

A: Some common examples of exponential functions include:

• $f(x) = 2^x$
• $f(x) = 3^x$
• $f(x) = 5^x$
• $f(x) = e^x$

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

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. A power function, on the other hand, is a function of the form $f(x) = x^n$, where $n$ is a real number. While both functions involve powers, the base of an exponential function is a constant, whereas the base of a power function is the variable.

Q: How do I simplify an exponential function?

A: To simplify an exponential function, you can use the following rules:

• $\left(a^m\right)^n = a^{mn}$
• $a^m \cdot a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$

Q: What is the inverse of an exponential function?

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

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use the following steps:

1. Determine the base and exponent of the function.
2. Plot a few points on the graph using the function.
3. Use the points to draw a smooth curve that represents the function.

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

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

• Population growth and decline
• Compound interest and finance
• Radioactive decay and nuclear physics
• Epidemiology and disease modeling

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the following steps:

1. Isolate the exponential term on one side of the equation.
2. Use the properties of exponents to simplify the equation.
3. Solve for the variable.

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

A: An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive real number and $x$ is the variable. A linear function, on the other hand, is a function of the form $f(x) = mx + b$, where $m$ and $b$ are real numbers. While both functions involve variables, the growth rate of an exponential function is not constant, whereas the growth rate of a linear function is constant.

Q: How do I determine if a function is exponential or linear?

A: To determine if a function is exponential or linear, you can use the following steps:

1. Check if the function is of the form $f(x) = a^x$.
2. Check if the function is of the form $f(x) = mx + b$.
3. If the function is of the form $f(x) = a^x$, it is an exponential function. If it is of the form $f(x) = mx + b$, it is a linear function.

Q: What are some common mistakes to avoid when working with exponential functions?

A: Some common mistakes to avoid when working with exponential functions include:

• Confusing the base and exponent of an exponential function.
• Failing to simplify an exponential function using the properties of exponents.
• Using the wrong rules for simplifying exponential functions.
• Failing to check the domain and range of an exponential function.