Problem 81 - Algebra and Trigonometry

Question

\(75-82\) . Determine whether the function \(f\) is even, odd, or neither. If \(f\) is even or odd, use symmetry to sketch its graph. $$ (x)=1-\sqrt[3]{x} $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand Even and Odd Functions

A function is even if \( f(-x) = f(x) \) for all \( x \). It is odd if \( f(-x) = -f(x) \) for all \( x \). If neither condition is satisfied, the function is neither even nor odd.

Step 2: Calculate \( f(-x) \)

Start with the given function \( f(x) = 1 - \sqrt[3]{x} \). Calculate \( f(-x) = 1 - \sqrt[3]{-x} \).

Step 3: Compare \( f(x) \) and \( f(-x) \) to check for evenness

Compare \( f(x) = 1 - \sqrt[3]{x} \) and \( f(-x) = 1 - \sqrt[3]{-x} \). They are not equal, so the function is not even.

Step 4: Check for oddness

Check if \( f(-x) = -f(x) \):\[ f(-x) = 1 - \sqrt[3]{-x} eq -(1 - \sqrt[3]{x}) \]Thus, \( f(-x) eq -f(x) \), so the function is not odd.

Step 5: Conclusion on Evenness/Oddness

Since \( f(x) \) is neither even nor odd, it does not possess symmetry about the y-axis or origin.

Step 6: Graph the Function

Since the function is neither even nor odd, use individual points from the function equation to sketch the graph. Plot points such as for \( x = 0, 1, -1, 8, -8 \), to understand the shape of the curve.

Key Concepts

Function Symmetry

Functions can exhibit symmetry, which helps in understanding their geometric representation. When a function is "even," it satisfies the condition: - \( f(-x) = f(x) \) for all values of \( x \). This means the graph is symmetrical about the y-axis. Think of it like folding the graph over the y-axis; if both sides match, the function is even. Examples include \( x^2 \) and \( \cos(x) \).
For functions that are "odd," the rule changes to:- \( f(-x) = -f(x) \). This symmetry revolves around the origin, like a 180-degree rotation. When you flip the graph over both axes, an odd function will perfectly align with itself.
In the exercise, neither condition was met for \( f(x) = 1 - \sqrt[3]{x} \). Therefore, the function does not exhibit symmetry about the y-axis or the origin. This means we cannot easily predict its shape based solely on symmetry properties.

Graph Sketching

When sketching graphs, symmetry can be a helpful guide. However, since \( f(x) = 1 - \sqrt[3]{x} \) is neither even nor odd, each point needs to be plotted independently. To graph this function:- Choose points like \( x = 0, 1, -1, 8, -8 \).- Calculate the value of the function at these points.- Plot these coordinates onto your graph.
For example, at \( x = 0 \), \( f(0) = 1 - \sqrt[3]{0} = 1 \). This tells us the line crosses the vertical line at \( y = 1 \). By plotting multiple points, you'll get a better picture of the function's behavior. This function has a distinctive curve that moves downward as \( x \) increases.
Graph sketching without symmetry relies heavily on observation from these points and connecting them with a smooth curve that reflects the function's changing rate.

Function Analysis

Analyzing the function \( f(x) = 1 - \sqrt[3]{x} \) involves understanding how it behaves across the domain of x values. Since this function is neither even nor odd, we analyze it from a general perspective. Considerations for this function include:- **Domain**: This function is defined for all real numbers because cube roots and constant operations are permissible across the real number line.- **Range**: Given that the cube root function shifts vertically by 1 unit, the range also spans all real numbers.- **Intercepts**: The y-intercept is easily found at point \( (0, 1) \).
Additionally, we examine any unique features such as asymptotic behavior and any points of inflection. Though asymptotic behavior is not present in \( 1 - \sqrt[3]{x} \), the graph decreases as \( x \) goes into both extremes. This knowledge makes it easier to draw accurate graphs and understand the overall trend.