Problem 48 - Elementary and Intermediate Algebra

Question

For each of the following functions, first sketch the graph of its associated function, \(f(x)=x^{2}, f(x)=x^{3},\) or \(f(x)=|x|\) Then draw the graph of function \(g\) using a translation and give its domain and range. See Examples 5 and \(6 .\) $$ g(x)=(x-1)^{3} $$

Step-by-Step Solution

Answer

The graph of \(g(x) = (x-1)^3\) is a horizontal shift of \(f(x) = x^3\) to the right by 1 unit. Its domain and range are both \((-infty, infty)\).

Step 1: Review of the Base Function

The base function we start with is \(f(x) = x^3\). This is a cubic function known for its characteristic S-shaped curve. It passes through the origin (0,0) and is symmetric about the origin.

Step 2: Sketch the Graph of f(x) = x^3

To sketch \(f(x) = x^3\), we plot key points: \((-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)\). The graph increases from left to right, passing through the origin with a steep incline. It has rotational symmetry about the origin.

Step 3: Identify the Transformation for g(x) = (x-1)^3

The function \(g(x) = (x-1)^3\) is a horizontal translation of \(f(x) = x^3\) to the right by 1 unit. This transformation is due to the \((x-1)\) in \(g(x)\), which shifts the graph horizontally.

Step 4: Sketch the Graph of g(x) = (x-1)^3

Plot the transformed points: \((-1, -8), (0, -1), (1, 0), (2, 1), (3, 8)\). The graph is similar in shape to \(f(x) = x^3\), but shifted right by 1 unit.

Step 5: Determine the Domain of g(x)

The domain of \(g(x) = (x-1)^3\) is all real numbers, since you can input any value of \(x\) and find a corresponding \(g(x)\). Expressed in interval notation, it is \((-infty, infty)\).

Step 6: Determine the Range of g(x)

The range of \(g(x) = (x-1)^3\) is also all real numbers. This is because the function can produce any output value over the real number line. It can also be expressed as \((-infty, infty)\).

Key Concepts

Translation of Functions

In mathematics, a translation of a function involves moving the graph of this function without changing its shape. This is typically done either horizontally or vertically. Let's focus on the horizontal translation.

• **Horizontal translation** involves shifting the graph left or right by a certain number of units. This transformation is reflected in changes made inside the function's argument with a formula like \(x - c\) or \(x + c\).
• When you see \(g(x) = (x-1)^3\), it implies that the original function \(f(x) = x^3\) is shifted 1 unit to the right. This is because the function is transformed through \((x-1)\).
• This transformation does not alter the function's domain or range or change its overall shape; the curve simply repositions.

Understanding translations helps in visualizing how functions behave under different transformations. It is essential for recognizing patterns in function shifts and quickly identifying the impact of different transformations on the graph of the function.

Cubic Functions

Cubic functions are mathematical functions characterized by the formula \(f(x) = x^3\). These functions are known for their distinctive S-shaped graph and are identifiable by a few particular features.

• **Shape:** The graph of a cubic function moves upward to the right and downward to the left, creating an S-shape. This curve shows rotational symmetry around the origin, often passing through points like (0,0), (1,1), and (-1,-1).
• **Behavior:** As \(x\) increases or decreases without bound, \(f(x)\) does the same. Meaning, as \(x\) approaches infinity or negative infinity, so does \(f(x)\), in their respective directions.
• **Graphical Features:** Typical key points for a cubic function include points where \(x\) plays a simple role in the equation. For \(y = x^3\), these are (-2, -8), (-1, -1), (0, 0), (1, 1), and (2, 8).

Cubic functions are fundamental in algebra and serve as a basis for understanding more complex polynomial functions. Their curves are crucial for modeling real-world phenomena where such patterns occur.

Domain and Range

The domain and range are critical concepts in understanding functions. They describe what inputs (domain) are acceptable and what outputs (range) you can expect from the function.

• **Domain:** The domain of a function is all the possible input values \(x\) can take. For a straightforward cubic function like \(g(x) = (x-1)^3\), there are no restrictions on \(x\), so the domain is all real numbers, expressed as \((-\infty, \infty)\).
• **Range:** The range is all the output values that the function can produce. For \(g(x) = (x-1)^3\), just like \(f(x) = x^3\), the outputs can be any real number. Thus, the range is also \((-\infty, \infty)\).This means there's no bound to the function's output values vertically on the graph.

Understanding these concepts is vital, particularly in ensuring functions are appropriately utilized in various mathematical contexts and in modeling situations. Knowing the domain and range helps identify the limitations and possibilities of given functions.