Problem 53 - Algebra A Combined Function

Question

Is the given function an exponential function? $$ f(x)=1.5 x^{2} $$

Step-by-Step Solution

Answer

The function \( f(x) = 1.5x^2 \) is not an exponential function.

Step 1: Understand the Definition of an Exponential Function

An exponential function is of the form \( f(x) = a \, b^x \) where \( a \) is a constant, \( b \) is a positive real number that is not equal to 1, and \( x \) is the exponent.

Step 2: Analyze the Given Function

The given function is \( f(x) = 1.5x^2 \). In this function, \( x \) is the base and \( 2 \) is the exponent.

Step 3: Compare with Exponential Function Form

In an exponential function, the variable \( x \) should be the exponent. In the function \( f(x) = 1.5x^2 \), \( x \) is not the exponent, it is the base.

Step 4: Conclusion on the Function Type

Since the form \( 1.5x^2 \) does not match the necessary form \( a \, b^x \), where \( x \) should be the exponent, the function is not an exponential function.

Key Concepts

Algebraic Functions

Algebraic functions are a class of functions that include any equation which can be constructed using algebraic operations, such as addition, subtraction, multiplication, division, and taking roots. One common example is a polynomial function. These functions are characterized by their familiar form, which often includes variables raised to integer powers.
When analyzing algebraic functions, it is important to understand the role and position of the variables involved. For instance:

• A linear function can be expressed as \( f(x) = ax + b \), where \( a \) and \( b \) are constants.
• A quadratic function takes the form \( f(x) = ax^2 + bx + c \), with \( a, b, \) and \( c \) as constants, and \( x \) as the base raised to the power of 2.
• Higher degree polynomial functions include terms like \( ax^3 \), \( ax^4 \), and so forth.

The function \( f(x) = 1.5x^2 \) is a quadratic function, which is a subtype of algebraic functions, specifically a polynomial of degree 2.

Mathematical Definitions

Understanding mathematical definitions is crucial in identifying whether a function is exponential or algebraic. An exponential function is specifically defined as having the variable in the exponent of the formula. The general form is \( f(x) = a \, b^x \), where:

• \( a \) is a constant multiplier,
• \( b \) is the base and must be a positive real number not equal to 1,
• \( x \) is the variable, which is located in the exponent position.

In contrast, polynomial functions have the variable as the base raised to an integer exponent. The difference between these definitions plays a vital role in function classification. Hence:

• \( 1.5x^2 \) is not exponential because \( x \) is the base, not the exponent.
• By understanding these definitions, we can correctly categorize \( 1.5x^2 \) as a polynomial rather than an exponential function.

Function Analysis

Function analysis involves examining the structure and behavior of a function to determine its type and characteristics. This process can include comparing the given function to known forms, like the exponential or polynomial forms.
Analyzing \( f(x) = 1.5x^2 \) critically involves determining its base and exponent. By inspecting the expression, we see:

• The variable \( x \) is the base, indicating an algebraic structure rather than an exponential one.
• The exponent is a constant, 2, which defines it as a quadratic form.

Function analysis helps in understanding how variables and constants interact in a formula, providing insight into behavior, such as growth rates and curve shapes. For quadratic functions:

• The graph is a parabola, opening upwards when the coefficient of \( x^2 \) is positive, as is the case here with \( 1.5 \).
• The vertex and axis of symmetry can be identified using standard quadratic formulas, but these are not necessary to determine the basic function type.

In summary, through careful function analysis, we can confirm that \( f(x) = 1.5x^2 \) does not fit the exponential category, due to its polynomial characteristics.