Problem 92 - Algebra A Combined Function

Question

Identify the domain and then graph each function. $$ f(x)=\sqrt[3]{x}-2 $$

Step-by-Step Solution

Answer

The domain is all real numbers, \( (-\infty, \infty) \). The graph is a shifted cube root function, moved 2 units down.

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt[3]{x} - 2 \). This is a cube root function with a vertical translation down by 2 units. The cube root function, \( \sqrt[3]{x} \), is defined for all real numbers since every real number has a cube root.

Step 2: Identify the Domain

The domain of \( f(x) = \sqrt[3]{x} - 2 \) is the set of all real numbers because the cube root function, \( \sqrt[3]{x} \), can accept any real number as input. Therefore, the domain is \( (-\infty, \infty) \).

Step 3: Graphing the Function

When graphing \( f(x) = \sqrt[3]{x} - 2 \), start by taking the graph of \( y = \sqrt[3]{x} \), which has a point at the origin (0,0) and passes through points such as (1,1) and (-1,-1). Shift this entire graph 2 units downwards to account for the \(-2\) in the function. This results in a graph that passes through (0,-2), (1,-1), and (-1,-3).

Key Concepts

Cube Root Function

The cube root function, expressed as \( \sqrt[3]{x} \), is a fundamental concept in mathematics. It is unique because it is defined for all real numbers. This is different from square root functions, which require non-negative inputs. Every real number, whether positive, negative, or zero, has a cube root.

• Positive numbers have positive cube roots because multiplying three positive numbers yields a positive product.
• Similarly, negative numbers have negative cube roots because the product of three negative numbers is also negative.
• Zero has a cube root of zero, as it is the only number that gives a product of zero when multiplied by itself three times.

Understanding this property of the cube root function allows you to immediately determine that its domain is all real numbers, represented as \((-\infty, \infty)\). This means you can input any real number into the function, and it will produce a real number output.

Vertical Translation

Vertical translation refers to shifting a function's graph up or down on the Cartesian plane. When we talk about translating a function vertically, we modify the y-coordinates of all its points while keeping the x-coordinates unchanged.In the function \( f(x) = \sqrt[3]{x} - 2 \), the \(-2\) indicates a vertical translation of the graph downwards by 2 units. Here's how this happens:

• A point on the original graph \( y = \sqrt[3]{x} \), such as \((1, 1)\), will move to \((1, -1)\) on \( f(x) = \sqrt[3]{x} - 2 \).
• Similarly, the origin point \((0, 0)\) on the cube root function shifts to \((0, -2)\).

The entirety of the function shifts downwards, maintaining the shape of the cube root graph, except that now every point is directly beneath where it started, by precisely two units.

Graphing Functions

Graphing functions is a process that helps us visualize the relationship between variables in an equation. To graph the function \( f(x) = \sqrt[3]{x} - 2 \), it's important to first visualize the parent function, \( y = \sqrt[3]{x} \).

• The parent graph starts at the origin, \((0,0)\), and for every unit increase or decrease in \(x\), the corresponding \(y\)-value increases or decreases as the cube root of \(x\).
• Main points to consider on this graph include \((-1, -1)\), which comes below the trend, and \((1, 1)\), which rises above it.

After plotting the parent function, apply the vertical translation. To account for the \(-2\) in \( f(x) = \sqrt[3]{x} - 2 \), you simply move every point on this basic graph 2 units downward. The new key points become \((0,-2)\), \((1,-1)\), and \((-1,-3)\), reflecting the vertical shift. This method of shifting helps ensure that you graph accurately without altering the function's inherent shape.