Question

Determine whether the equation defines y as a function of \(x .\) \(y=x^{3}\)

Short answer

Yes, y = x^{3} defines y as a function of x.

Step-by-step solution

Understand the Problem

To determine whether the equation defines y as a function of x, one must know the definition of a function. In a function, each input value (x) should map to exactly one output value (y).

Analyze the Given Equation

Examine the given equation: y = x^{3}This suggests that y is determined by raising x to the power of 3.

Determine the Behavior of the Equation

Evaluate whether each value of x produces exactly one value of y. For any real number x, raising it to the power of 3 will always produce a unique real number y.

Conclude

Since for every input x there is exactly one output y, the equation y = x^{3} defines y as a function of x.

Key concepts

Definition of a Function

Understanding the definition of a function is crucial in identifying whether a given equation represents a function. A function is a relationship that assigns exactly one output to each input. To put it simply, if you put a value into the function (input), you will get a single specific value out of it (output). In mathematical terms, if every x-value (input) corresponds to exactly one y-value (output), then you're dealing with a function.

For example, in the equation y = x^3, you plug in an x-value, perform the operation (raising it to the power of 3), and get a unique y-value. This relationship is consistent and reliable, meaning the equation defines y as a function of x.

So, remember: a function has only one output for each input. This characteristic helps you determine whether a given equation is indeed a function.

Unique Output

When evaluating whether an equation defines a function, it's important to ensure that every input has a unique output. In mathematical terms, this means that for any x-value in the domain, there is one and only one y-value.

Consider the equation y = x^3. For any given real number x, raising it to the power of 3 will always result in a unique real number y. There are no exceptions to this rule for the equation given. This property satisfies the condition of a function.

To break it down further:

• If x = 2, then y = 2^3 = 8.
• If x = -1, then y = (-1)^3 = -1.
• If x = 0, then y = 0^3 = 0.

Each x-value results in only one distinct y-value, reinforcing the idea that the equation y = x^3 defines y as a function of x. This pattern confirms that each input leads to a unique output, a hallmark of functions.

Function Behavior

Analyzing the behavior of a function allows you to understand how the output changes with different inputs. For the equation y = x^3, there's a consistent method to determine the output for any input.

By examining the behavior of the function y = x^3, we observe:

• As x increases, y increases since x^3 gets larger.
• As x decreases, y also decreases because raising a negative number to the power of 3 results in a negative number.
• If x is 0, y is also 0, because any number raised to the power of 3 is still 0.

These observations show a smooth, continuous relationship between the input x and the output y. There are no jumps or breaks in the graph of y = x^3, further indicating it's a function.

This behavior helps us conclude that the function y = x^3 consistently maps each x-value to one unique y-value. This analysis solidifies our understanding of the function's characteristics and confirms its function status.