Problem 2 - Algebra 2 and Trigonometry

Question

Explain why \(y=b^{x}\) is not an exponential function for \(b=1\)

Step-by-Step Solution

Answer

\(y = b^x\) is not exponential for \(b=1\) because it simplifies to a constant function, \(y=1\).

Step 1: Define an Exponential Function

An exponential function is generally defined as a function of the form \(y = b^x\), where \(b\) is a positive constant other than 1, and \(x\) is the variable exponent.

Step 2: Consider the Case When \(b = 1\)

Substitute \(b = 1\) into the function to get \(y = 1^x\). For any value of \(x\), \(1^x = 1\). Thus, the function simplifies to \(y = 1\).

Step 3: Evaluate the Nature of the Function

The function \(y = 1\) is a constant function, not an exponential function. Exponential functions must vary with \(x\), but in this case, \(y\) remains the same for every \(x\).

Step 4: Confirm the Functional Behavior

For a function to be classified as exponential, it should exhibit exponential growth or decay, meaning the output should change exponentially as \(x\) changes. Since \(y = 1\) does not vary with \(x\), it does not meet this criterion.

Key Concepts

Constant Function

A constant function is a simple yet important concept in mathematics. It is a function where the output value remains the same regardless of the input. This means that no matter what value you choose for the variable, the result is always the same constant number. Constant functions can be written in the form of:

• \( y = c \)
• where \( c \) is a constant.

In the case of the given problem, when \( b = 1 \) in the function \( y = b^x \), the function becomes \( y = 1^x \), which simplifies to \( y = 1 \). This is a constant function because the output is always 1, no matter what the value of \( x \) is.

Whenever you see a graph of a constant function, it looks like a horizontal line, demonstrating that the value does not change. Constant functions serve as the baseline to compare other functions that do change with different inputs.

Exponential Growth

Exponential growth is a fundamental characteristic of exponential functions. When we talk about exponential growth, we are describing a situation where the variable increases at a rate proportional to its current value. This is typical for processes like population growth or compound interest.

For a function \( y = b^x \), exponential growth occurs when the base \( b \) is greater than 1. In this scenario, as \( x \) increases, \( y \) increases rapidly. This is because each increment in \( x \) multiplies \( y \) by another factor of \( b \).

• For example, if \( b = 2 \), then \( y = 2^x \) grows rapidly as \( x \) increases: doubling with each step.

Exponential growth contrasts with linear growth, where quantities increase by the same amount (not factor) with each step. In practical terms, exponential growth can lead to large numbers quickly, which is why it is often used to model fast-changing situations.

Variable Exponent

Having a variable exponent can notably alter the behavior of a function. In exponential functions, the variable exponent is typically represented as \( x \). It plays a crucial role in determining how the function behaves.

Exponential functions are generally defined as \( y = b^x \), where \( b \) is the positive constant base and \( x \) is the exponent. The variable exponent \( x \) allows the exponent to vary, which creates a dynamic change in the resultant value as \( x \) changes.

• If \( b > 1 \), as \( x \) increases, the function grows rapidly.
• If \( 0 < b < 1 \), as \( x \) increases, the function approaches zero and experiences exponential decay.

In the case of a constant function, where \( b = 1 \), the variable exponent still exists, but it does not influence the output, since \( 1^x = 1 \). Thus, the function does not exhibit typical exponential behavior, emphasizing the importance of the base in these functions.

Positive Constant

In the context of exponential functions, a positive constant is crucial for defining their nature. When we talk about the constant \( b \) in the function \( y = b^x \), it must be positive and cannot be equal to 1 to maintain true exponential form.

• Why it must be positive: If \( b \) were negative, the function could result in non-real values for certain exponents \( x \), making it unpredictable.
• Why it shouldn't be 1: If \( b = 1 \), the function simplifies to a constant function, as discussed before, which lacks the dynamic quality essential for exponential functions.

The role of the positive constant is to ensure the consistent proportional scaling that defines exponential behavior, either growth or decay. The choice of \( b \) directly affects the speed at which this occurs and the overall nature of the function. Thus, selecting an appropriate positive constant is key when modeling real-world situations with exponential functions.