Problem 83 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. \(f(x)=1^{x}\) is not an exponential function.

Step-by-Step Solution

Answer

The statement \(f(x)=1^{x}\) is not an exponential function is true because here, \(f(x)\) is a constant function, not an exponential function.

Step 1: Analyze the given function

The given function is \(f(x) = 1^{x}\). In this function, the base of the exponent (i.e., the number being raised to the power of \(x\)) is one.

Step 2: Compare with the definition of an exponential function

An exponential function is a function of the form \(f(x) = b^{x}\) where \(b\) (the base of the exponent) is a constant greater than zero. Here, \(b = 1\), which is allowed by the definition of the exponential function. However, there is an important distinction that \(b\) is not equal to one in an exponential function. Here, the base of the exponent is one, which is not allowed in an exponential function.

Step 3: Identify the type of the function

\(\forall x \in \mathbb{R}, f(x)=1^{x}=1\). Therefore, \(f(x)\) is a constant function, not an exponential function.

Key Concepts

Exponential Function Definition

An exponential function plays a crucial role in various mathematical and practical applications. It is uniquely characterized by its form, where a constant base is raised to the power of a variable. Specifically, an exponential function is written as
\( f(x) = b^x \),
where \( b \) is the base that is a positive real number other than 1, and \( x \) is the exponent that is any real number. This precise stipulation— that \( b \) must be positive and not equal to 1— distinguishes an exponential function from other types of functions.
The function demonstrates a rapid growth or decay rate, depending on whether the base is greater than 1 or between 0 and 1, respectively. The value of the function multiplies by a fixed factor whenever the exponent increases by 1, exhibiting a consistent proportional change that is the hallmark of exponential behavior.
Understanding the exponential function is essential for studying compound interest in finance, population growth in biology, and radioactive decay in physics, among other applications.

Constant Function

In contrast to the dynamic nature of exponential functions, a constant function represents the notion of mathematical consistency and simplicity. A constant function is defined as
\( f(x) = c \),
where \( c \) is a fixed real number, and no matter the input value for \( x \), the function's output remains unchanged.
This type of function results in a straight horizontal line when graphed on the Cartesian plane, indicating that for all inputs, the output is consistently the same value, \( c \). In the textbook example, \( f(x) = 1^x \) simplifies to \( f(x) = 1 \) for all real numbers \( x \), which means it's indeed a constant function and not an exponential function. Constant functions are vital in equations and models where a value needs to remain unchanged across different contexts or conditions.

Properties of Exponents

Understanding the properties of exponents is fundamental for working with exponential functions. These rules govern how terms with exponents are manipulated and simplified. Here are key properties that are frequently utilized:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \) — when multiplying like bases, add their exponents.
• Power of a Power: \( (a^m)^n = a^{m \cdot n} \) — when raising a power to another power, multiply the exponents.
• Power of a Product: \( (ab)^n = a^n \cdot b^n \) — when raising a product to a power, raise each factor to the power separately.
• Quotient of Powers: \( a^m / a^n = a^{m-n} \) — when dividing like bases, subtract the exponent of the denominator from the exponent of the numerator.
• Zero Exponent: \( a^0 = 1 \) — any non-zero base raised to the power of zero equals one.
• Negative Exponent: \( a^{-n} = 1/a^n \) — a negative exponent indicates the reciprocal of the base raised to the positive exponent.

These properties are the building blocks for more complex operations involving exponents and are crucial for simplifying expressions, solving equations, and understanding the behavior of exponential functions.