Question

In Exercises 5-12, (a) identify the domain and range and (b) sketch the graph of the function. $$y=\sqrt[4]{-x}$$

Short answer

The domain of the function \(y = \sqrt[4]{-x}\) is \(-\infty, 0]\) and the range is \([0, \infty)\). The graph of the function starts at the origin and extends upwards to the left into the second quadrant.

Step-by-step solution

Identify the Domain

For the function \(y = \sqrt[4]{-x}\), it is understood that the radicand \(-x\) has to be greater than or equal to zero, because the fourth root of a negative number isn't a real number. Therefore, this condition gives us \(x \leq 0\). Hence, the domain of the function is \(-\infty, 0]\).

Identify the Range

Considering that \(y = \sqrt[4]{-x}\) is a real valued function, output \(y\) of the function should always be a real number. The fourth root of any nonnegative number is nonnegative, hence \(y\) lies in the range \([0, \infty)\).

Sketch the Graph

To construct the graph, plot points that satisfy the function equation, and make note that as \(x\) gets lesser in negative, \(y\) increases. The graph starts at the origin (0,0) and extends upwards as you go left in the negative x-direction. The graph will only exist in the second quadrant because \(x\) is less than or equal to 0 and \(y\) is greater than or equal to 0.

Key concepts

Domain of a Function

Understanding the domain of a function is essential for graphing and analyzing its behavior. Simply put, the domain encompasses all the possible input values (usually x-values) for which the function is defined and produces real numbers as outputs. In mathematical terms, it describes the set of all possible values of the independent variable.

For the function in question, given by the expression \(y = \frac[4]{-x}\), we need to consider which values of \(x\) make the expression inside the fourth root, known as the radicand, a valid input. Since taking any even root of a negative number does not yield a real number, the domain in this case is limited to non-positive numbers, leading to the domain being \(x \leq 0\). This means the function only accepts zero and negative values for \(x\) as legitimate inputs, providing a clear boundary that the function does not cross.

Range of a Function

The range of a function, in contrast to the domain, represents the set of all possible output values (usually y-values) that a function can produce. Determining the range is essentially about identifying the limits within which the function’s output values lie.

For the fourth root function \(y = \frac[4]{-x}\), since the input to the root function can only be zero or positive (as the domain restricts \(x\) to be non-positive), the outputs are also constrained to be non-negative. The fourth root of zero is zero, and the fourth root of any positive number is a positive real number. Therefore, the range of this function is all non-negative real numbers, or \([0, \infty)\).

Real-Valued Functions

Real-valued functions are those functions that have real numbers as inputs (domain) and produce real numbers as outputs (range). Most functions in elementary algebra and calculus are real-valued functions. These functions are fundamental because they can model real-world phenomena where both the quantities we measure (inputs) and the resulting observations (outputs) are real numbers.

The function \(y = \frac[4]{-x}\) is a real-valued function as it accepts real numbers as inputs and produces real numbers as outputs, under the condition that the inputs adhere to the function’s domain. When sketching or analyzing real-valued functions, it’s crucial to ensure that the operations within the function (like roots in this case) remain within the realm of real numbers.

Fourth Root Properties

The fourth root of a number is a value that, when raised to the power of four, equals the original number. One key property of the fourth root, especially relevant to the function \(y = \frac[4]{-x}\), is that it only yields real numbers when the radicand is non-negative. Therefore, the input must be greater than or equal to zero for the fourth root to be defined within the realm of real numbers.

Another property is that the fourth root of a number is always non-negative if the number itself is non-negative. This is because raising a positive number or zero to any even exponent results in a non-negative number. These properties are crucial in identifying the range of the function \(y = \frac[4]{-x}\) and in understanding why the function graph only occupies the second quadrant, where \(x \leq 0\) and \(y \geq 0\).