Question

Consider the function \(f(x)=2 x-3 .\) What is the range of the absolute value of this function? (A) \(y<-3\) (B) \(y \leq 0\) (C) \(y \geq 0\) (D) \(y>5\)

Short answer

The range is \( y \geq 0 \). This corresponds to option (C).

Step-by-step solution

Consider the Function

First, note that the given function is \( f(x) = 2x - 3 \). We are asked to determine the range of the absolute value of this function, which is \( |f(x)| = |2x - 3| \). The absolute value indicates that any negative output of the function will be converted to a positive value.

Analyze the Absolute Value Function

The absolute value expression \( |2x - 3| \) will be zero or greater, i.e., \( |2x - 3| \geq 0 \), because absolute values are never negative. This expression reaches its minimum value of 0 when \( 2x - 3 = 0 \) (i.e., when \( x = \frac{3}{2} \)).

Determine the Range

Because the absolute value of any real number, including linear expressions, covers all non-negative real numbers, the range of \( |f(x)| = |2x - 3| \) is \( y \geq 0 \). This option corresponds to the possible non-negative outputs generated by any real value input into the linear expression.

Key concepts

Absolute Value

Absolute value is a mathematical operation that takes a real number and returns its non-negative value. It turns negative numbers into positive ones while leaving positive numbers and zero unchanged. The absolute value of a number \( a \) is symbolized as \( |a| \). This concept is widely used because it indicates the distance of a number from zero on a number line, irrespective of direction.

For example, the absolute value of -5 is \( | -5 | = 5 \), and for 3, it is \( |3| = 3 \). In the context of functions, like \( f(x) = 2x - 3 \), the absolute value operation \( |f(x)| \) ensures that outputs are always non-negative. No matter what value \( x \) takes, \( |2x - 3| \) will always be zero or greater, never negative. Hence, we can confidently say that the range of the absolute value of a function will include zero and all positive values. This property is particularly important for determining the range of expressions involving absolute values.

Range of a Function

The range of a function is a fundamental concept in mathematics, representing all the possible output values a function can produce. Knowing the range helps understand the behavior of the function. For a given function \( f(x) \), the range consists of all \( y \) values that \( f(x) \) can output when \( x \) varies over all possible inputs.

In the provided exercise, we analyze \( |f(x)| = |2x - 3| \). Since absolute values can only result in zero or positive outputs, the range must be \( y \geq 0 \). This information tells us that no matter what input \( x \) we choose, the resulting output will never be negative.

Understanding the range of functions involving absolute values can be straightforward since absolute values inherently ensure outputs are never below zero. When tackling problems involving absolute values and their range, always remember that negative results are not possible.

Linear Functions

Linear functions are one of the simplest types of functions and are fundamental in math. A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These functions graph as straight lines, and each point on the line signifies a solution to the function.

In the problem we're examining, \( f(x) = 2x - 3 \) is a linear function. Here, \( m = 2 \) (the slope), and \( b = -3 \) (the y-intercept). The slope of a linear function indicates the steepness of the line, while the y-intercept shows where the line crosses the y-axis.

When working with linear functions, one key aspect to understand is how the graph's line affects the function's output. In combination with absolute values, like \(|2x - 3|\), linear functions become versatile in handling a wider range of real-world problems since they can model both direct and inverse relationships. Linear functions are always valued as they provide straightforward, predictable outcomes.