Problem 1 - Algebra and Trigonometry

Question

The function \(f(x)=5^{x}\) is an exponential function with base _____; \(f(-2)=\) _____, \(f(0)=\) _____, \(f(2)=\) _____, and \(f(6)=\) _____.

Step-by-Step Solution

Answer

The base is 5; \(f(-2) = \frac{1}{25}\), \(f(0) = 1\), \(f(2) = 25\), \(f(6) = 15625\).

Step 1: Identify the Base of the Exponential Function

The function given is \(f(x) = 5^x\). In an exponential function of the form \(a^x\), the base is the number \(a\). Hence, for \(f(x) = 5^x\), the base is 5.

Step 2: Calculate \(f(-2)\)

Substitute \(-2\) for \(x\) in the function. So, \(f(-2) = 5^{-2}\). By the laws of exponents, \(5^{-2} = \frac{1}{5^2}\). Now calculate the power: \(5^2 = 25\). Thus, \(f(-2) = \frac{1}{25}\).

Step 3: Calculate \(f(0)\)

Substitute 0 for \(x\) in the function. Thus, \(f(0) = 5^0\). According to the property of exponents, any non-zero number raised to the power of zero equals 1. Therefore, \(f(0) = 1\).

Step 4: Calculate \(f(2)\)

Substitute 2 for \(x\) in the function. Thus, \(f(2) = 5^2\). Calculate the power: \(5^2 = 25\). Therefore, \(f(2) = 25\).

Step 5: Calculate \(f(6)\)

Substitute 6 for \(x\) in the function. Thus, \(f(6) = 5^6\). Now, calculate the power: \(5^6 = 15625\). Hence, \(f(6) = 15625\).

Key Concepts

Understanding the Base of Exponential Functions

In an exponential function, such as the one in the exercise, the base is a crucial component. The function given is \(f(x) = 5^x\). Here, the "base" is the number that is repeatedly multiplied by itself, corresponding to the value of \(x\), known as the exponent. This base must always be greater than zero for the function to be defined.The base of an exponential function affects the growth or decay of the function:

• If the base is greater than 1, the function represents exponential growth, meaning it will increase rapidly as \(x\) gets larger.
• If the base is between 0 and 1, it represents exponential decay, where the function will decrease and approach zero as \(x\) increases.

For our function \(f(x) = 5^x\), the base is 5, indicating that with each increase in \(x\), the function will grow steeply.

Applying the Laws of Exponents

The laws of exponents provide a set of rules that simplify calculations involving exponential expressions. These rules are essential when working with functions like \(f(x) = 5^x\).Here are some key laws used in the solution:

• Product of Powers: When multiplying like bases, you add the exponents, \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing like bases, subtract the exponents, \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: When raising a power to another power, multiply the exponents, \((a^m)^n = a^{mn}\).
• Negative Exponent: A negative exponent indicates the reciprocal of the positive exponent, \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponent: Any non-zero base raised to the power of zero equals 1, \(a^0 = 1\).

In the exercise, when you calculate \(f(-2) = 5^{-2}\), the negative exponent law tells us to take the reciprocal, resulting in \(\frac{1}{5^2}\). Similarly, for \(f(0) = 5^0\), the zero exponent law gives us a result of 1.

Steps to Calculate Exponents

Calculating exponents involves applying the base and exponent to derive a result. It's straightforward once you understand the process and rules.Here's a simple guide to calculating exponents as shown in the exercise:

• Identify the base and the exponent. For example, in \(5^2\), 5 is the base and 2 is the exponent.
• Multiply the base by itself as many times as the exponent indicates. So, \(5^2 = 5 \times 5\).
• Simplify the expression to get the result. Here, \(5 \times 5 = 25\).

In the exercise, when calculating \(f(6) = 5^6\), you multiply 5 by itself six times, resulting in a large number, 15625. This method is critical to solving and understanding exponential functions efficiently. Practice these steps to become more comfortable with finding powers of numbers.