Problem 54 - Algebra A Combined Function

Question

Is the given function an exponential function? $$ g(x)=3^{x} $$

Step-by-Step Solution

Answer

Yes, \( g(x) = 3^x \) is an exponential function.

Step 1: Define an Exponential Function

An exponential function has the form \( f(x) = a \cdot b^x \), where \( a eq 0 \) and \( b > 0 \) and \( b eq 1 \). The variable \( x \) is in the exponent position.

Step 2: Analyze the Given Function Form

The function given is \( g(x) = 3^x \). Here, we identify that \( a = 1 \) and \( b = 3 \). The function is of the form \( a \cdot b^x \) with base \( b = 3 \), which is greater than 0 and not equal to 1.

Step 3: Conclusion

Since the given function \( g(x) = 3^x \) fits the form \( f(x) = a \cdot b^x \) with \( a = 1 \) and \( b = 3 \), it satisfies all criteria of an exponential function.

Key Concepts

Function Analysis

When analyzing any mathematical function, it's crucial to understand its form and behavior. A function analysis involves determining the type of function you have and assessing its characteristics. For exponential functions, they have a specific form which is crucial: \( f(x) = a \cdot b^x \). To understand whether a function is exponential, you should identify:

• The base \( b \): This must be a positive number and not equal to one.
• The coefficient \( a \): This can be any non-zero real number.
• The exponent: In exponential functions, the variable \( x \) is in the exponent position, which indicates the function changes its rate of increase or decrease exponentially as \( x \) changes.

By examining these components, you can conclude the nature of the function at hand. Hence, the function \( g(x) = 3^x \) is an exponential one as it fits this established form.

Base of Exponential Function

The base \( b \) in an exponential function is an essential component. It dictates how the function behaves as \( x \) varies. A few important things to remember about the base:

• The base \( b \) must be greater than zero. This ensures that the function grows or decays consistently.
• It is crucial that \( b eq 1 \), because if \( b \) equals 1, the function becomes constant rather than exponential.

In the function \( g(x) = 3^x \), the base is 3. The choice of base greatly impacts the growth rate of the function. If the base is greater than 1, the function grows over time. If the base is between 0 and 1, the function would decrease over time.

Exponent Position

The exponent in an exponential function is where the variable \( x \) resides. This is key because:

• The position of \( x \) in the exponent differentiates exponential functions from other types of functions, like polynomial or logarithmic functions.
• It affects how quickly the value of the function will change as \( x \) changes. A small change in \( x \) can lead to a large change in the function's output.

For the function \( g(x) = 3^x \), the variable \( x \) is indeed in the exponent position. As \( x \) increases, the value of \( g(x) \) increases exponentially due to this position. Understanding this allows you to predict and interpret the behavior of exponential functions effectively.