Linear Function Properties Determining Truth Of Statements

To tackle the question of if f(x) is a linear function, and which statement must be true, we first need to solidify our understanding of what a linear function truly is. In mathematics, a linear function is defined as a function whose graph produces a straight line. This fundamental characteristic dictates the form and properties that a linear function possesses. Linear functions are among the most basic yet crucial concepts in algebra and calculus, forming the foundation for more advanced mathematical ideas. At its core, a linear function is characterized by a constant rate of change, meaning that for every unit increase in the input variable (typically x), the output variable (typically f(x) or y) changes by a constant amount. This constant rate of change is what we commonly refer to as the slope of the line. The slope essentially quantifies the steepness and direction of the line, indicating how much the function's value increases or decreases for each unit change in the input. The representation of a linear function mathematically is typically expressed in the slope-intercept form, which is f(x) = mx + b. This form provides a clear and concise way to understand the function's behavior, where 'm' represents the slope and 'b' represents the y-intercept. The y-intercept is the point where the line crosses the y-axis, indicating the value of the function when the input variable x is zero. Understanding this standard form is essential for analyzing and manipulating linear functions, as it directly reveals the two key parameters that define the line: its slope and its y-intercept. Furthermore, the linearity of the function implies that it satisfies two critical properties: additivity and homogeneity. Additivity means that the function of the sum of two inputs is equal to the sum of the functions of the individual inputs, expressed mathematically as f(x + y) = f(x) + f(y). Homogeneity, on the other hand, means that scaling the input by a constant scales the output by the same constant, expressed as f(cx) = cf(x), where 'c' is a constant. These properties are fundamental to many mathematical operations and theorems involving linear functions. The graph of a linear function is always a straight line, a visual representation that underscores the constant rate of change. This straight line can have different slopes, ranging from horizontal (slope of zero) to vertical (undefined slope), and it can intersect the y-axis at any point, determined by the y-intercept. The absence of any curvature or bends in the graph is a direct consequence of the function's linearity. In contrast, non-linear functions, such as quadratic or exponential functions, exhibit curved graphs due to their varying rates of change. Understanding the characteristics of linear functions is not only essential for solving mathematical problems but also for applying these concepts in real-world scenarios. Linear functions are widely used to model relationships that exhibit a constant rate of change, such as the relationship between distance and time at a constant speed, or the relationship between cost and the number of items purchased at a fixed price per item. These applications highlight the practical significance of linear functions in various fields, including physics, economics, and engineering.

Given our understanding of linear functions, let's break down each option to determine which statement must be true.

Option A: f(x) has no constant term.

This statement is not necessarily true. A linear function can indeed have a constant term, which is represented by 'b' in the slope-intercept form f(x) = mx + b. The constant term 'b' represents the y-intercept of the line, indicating where the line crosses the y-axis. For instance, the function f(x) = 2x + 3 is a linear function with a constant term of 3. This constant term shifts the entire line vertically, but it does not affect the linearity of the function. Therefore, a linear function can exist with or without a constant term. A linear function that does have a constant term simply intersects the y-axis at a point other than the origin. To further illustrate this, consider the function f(x) = x - 5. Here, the constant term is -5, and the line intersects the y-axis at the point (0, -5). The presence of this constant term does not violate the linearity of the function; it merely adjusts the vertical position of the line on the coordinate plane. In contrast, if a linear function had no constant term (i.e., b = 0), it would pass through the origin (0, 0). For example, f(x) = 4x is a linear function with no constant term, and its graph is a straight line that goes through the origin. The crucial takeaway is that the presence or absence of a constant term does not define whether a function is linear; rather, it determines where the line intersects the y-axis. Therefore, a linear function can certainly have a constant term, making option A incorrect.

Option B: f(x) has no x²-term.

This statement is true. The defining characteristic of a linear function is that the highest power of the variable x is 1. An x²-term would make the function a quadratic function, which has a curved graph (a parabola) rather than a straight line. For example, the function f(x) = x² + 2x - 1 is a quadratic function due to the presence of the x² term, and its graph is a parabola. The inclusion of an x² term introduces a non-constant rate of change, which is a key differentiator between linear and quadratic functions. In a linear function, the rate of change (slope) is constant, meaning that the function increases or decreases by the same amount for each unit increase in x. However, in a quadratic function, the rate of change varies depending on the value of x, leading to the curved shape of the graph. This variation in the rate of change is a direct consequence of the x² term. If a function contains an x² term, it will exhibit a parabolic behavior rather than the linear behavior characteristic of linear functions. Therefore, the presence of an x² term fundamentally alters the nature of the function, making it non-linear. To further clarify, any term with a power of x greater than 1 (e.g., x³, x⁴, etc.) would also disqualify the function from being linear. The essence of linearity lies in the constant rate of change, which is ensured by having only x terms raised to the power of 1 or 0 (constant terms). Thus, the absence of an x² term is a necessary condition for a function to be linear. This makes option B the correct answer.

Option C: f(x) has no terms with a coefficient other than 1.

This statement is not necessarily true. The coefficient of the x-term in a linear function can be any real number (except zero, as that would eliminate the x-term altogether). For example, f(x) = 5x + 2 is a linear function where the coefficient of x is 5, which is not 1. The coefficient of x determines the slope of the line, and changing this coefficient changes the steepness of the line but does not affect its linearity. A coefficient other than 1 simply means that the rate of change (slope) is different from 1, but the function still maintains a constant rate of change, which is the hallmark of linear functions. To further illustrate, consider the functions f(x) = -3x + 1 and f(x) = 0.5x - 4. In the first function, the coefficient of x is -3, indicating a negative slope (the line decreases as x increases). In the second function, the coefficient is 0.5, indicating a gentler positive slope. Despite these varying coefficients, both functions are linear because they produce straight-line graphs. The coefficient can also be a fraction, a decimal, or any other real number, as long as it is multiplied by x to the power of 1. The key is that the relationship between x and f(x) remains linear, meaning that the change in f(x) is directly proportional to the change in x. Therefore, the statement that a linear function has no terms with a coefficient other than 1 is incorrect.

Option D: f(x) has no x-term.

This statement is also not true. A linear function, by definition, must have an x-term (unless the function is a constant function, where the slope is zero, but it's still considered a special case of a linear function). Without an x-term, the function would simply be a constant, f(x) = b, which represents a horizontal line. While a horizontal line is a straight line, and therefore a type of linear function, the general form of a linear function includes an x-term to account for non-horizontal lines. The x-term is what gives the function its slope, allowing it to increase or decrease as x changes. The absence of an x-term would mean that the function's value remains constant regardless of the value of x. To elaborate, consider the functions f(x) = 2x + 1 and f(x) = -x + 3. Both of these functions have x-terms, and their graphs are non-horizontal lines with different slopes. The presence of the x-term is crucial for capturing the linear relationship between x and f(x). In contrast, a function like f(x) = 5 has no x-term and represents a horizontal line at y = 5. This is a constant function, and while it is linear, it does not exemplify the general case of a linear function with a non-zero slope. Therefore, the statement that a linear function has no x-term is not universally true, as it would exclude many linear functions with varying slopes. The existence of the x-term is what allows linear functions to model relationships where the output changes linearly with the input, making it an essential component of the function's definition.

The correct answer is B. f(x) has no x²-term. This is the only statement that must be true for a function to be linear.