Question

Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed, distance \(=\)speed time elapsed.A function \(y=f(x)\) such that \(y\) is 1 less than the cube of \(x\).

Short answer

Answer: The formula of the function that describes y is 1 less than the cube of x is \(y = x^3 - 1\) and its domain is \((-\infty, \infty)\).

Step-by-step solution

Determine the formula of the function

We have to find the function \(y=f(x)\) such that \(y\) is 1 less than the cube of \(x\). Mathematically, it can be expressed as \(y=x^3 - 1\). This will be the equation of our function.

Graph the function

To graph the function \(y=f(x)=x^3 - 1\), we plot a few points corresponding to a few x-values. Some example points are: (-2, -9), (-1, -2), (0, -1), (1, 0), (2, 7). Using these points, we can draw a smooth curve representing the graph of our function \(y=f(x)=x^3 - 1\).

Determine the domain of the function

The domain of the function consists of all possible x-values for which the function is defined. Since we have no restrictions on the values of x for our function, it means that \(x\) can take any real value. Therefore, the domain of our function \(f(x)=x^3 - 1\) is: Domain: \((-\infty, \infty)\)

Key concepts

Function Formula

At the heart of understanding any mathematical function is the function formula, which acts as a blueprint for the relationship between variables. In our exercise, we identify a function formula that characterizes the connection between two quantities, distance and time, with the context hinting that the relationship is cubic. Specifically, the function formula is presented as

\(y = x^3 - 1\).

This compact notation is incredibly powerful. It tells us that for any input value \('x'\), the output \('y'\) is determined by taking the cube of \('x'\) and subtracting one. The simplicity of this formula belies its utility — by plugging in different \('x'\) values, we can see how \('y'\) behaves in response. This formula also sets the stage for graphing the function and exploring its domain. When learning about function formulas, students should remember that it's more than just an equation to memorize; it represents a general law that dictates how quantities in a function relate to each other.

Domain of a Function

The domain of a function is a fundamental concept that frames the set of inputs over which the function is defined. In other words, it answers the question: 'What are the possible \('x'\) values that I can put into my function formula?'

For the cubic function \(y = x^3 - 1\), as shown in the exercise's solution, the domain is all real numbers, which is denoted as

\(\text{Domain}: (-\infty, \infty)\).

This is because we can cube any real number and then subtract one without any mathematical complications such as division by zero or taking the square root of a negative number. Understanding the domain is crucial for graphing the function accurately and for solving real-world problems that the function might represent. Always make it a point to verify the domain in context; while the domain is all real numbers in pure mathematics, in applied situations, physical or practical constraints may limit the possible \('x'\) values.

Cubic Functions

Cubic functions are a category of polynomial functions characterized by the highest degree of the variable being three. The general form of a cubic function is \(y = ax^3 + bx^2 + cx + d\), where \('a'\), \('b'\), \('c'\), and \('d'\) are constants. The function \(y = x^3 - 1\) from our exercise is a specific example of a cubic function.

These types of functions are notable for their 'S' shaped curves which can be seen when graphed. This shape is due to the fact that as \('x'\) becomes large or small, the \('x^3'\) term starts to dominate the behavior of the function, resulting in steep ascents and descents on the graph. Students must understand that cubic functions can have one, two, or three x-intercepts and can turn at most twice; these are known as turning points. In the case of \(y = x^3 - 1\), the graph passes through the point (0, -1) and heads upwards to the right and downwards to the left, illustrating the classic behavior of a cubic function.