Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Based on the ordered pairs seen in each table, make a conjecture about whether the finction \(f\) is even, odd, or neither even nor odd. $$\begin{array}{r|r}x & f(x) \\\\-3 & 10 \\\\-2 & 5 \\\\-1 & 2 \\\0 & 0 \\\1 & -2 \\\2 & -5 \\\3 & -10\end{array}$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Recall Definitions of Even and Odd Functions

An even function satisfies the condition \(f(-x) = f(x)\) for all \(x\) in its domain. An odd function satisfies \(f(-x) = -f(x)\) for all \(x\) in its domain. If neither of these conditions hold for all \(x\), the function is neither even nor odd.

Step 2: Check Even Function Condition

Look at the table and check if \(f(-x) = f(x)\) for each ordered pair. - For \(x = 3\), \(f(-3) = 10\) and \(f(3) = -10\). This is not equal, so it does not satisfy the even condition.- For \(x = 2\), \(f(-2) = 5\) and \(f(2) = -5\). This is not equal.- Continue checking for \(x = 1\), the condition is still not satisfied.

Step 3: Check Odd Function Condition

Check if \(f(-x) = -f(x)\) for each ordered pair.- For \(x = 3\), \(f(-3) = 10\) and \(-f(3) = -(-10) = 10\). This condition holds.- For \(x = 2\), \(f(-2) = 5\) and \(-f(2) = -(-5) = 5\). This condition holds.- For \(x = 1\), \(f(-1) = 2\) and \(-f(1) = -(-2) = 2\). This condition still holds.- For \(x = 0\), \(f(-0) = 0\) and \(-f(0) = -0 = 0\). This is trivially satisfied.These checks show that the function satisfies odd conditions for all given \(x\).

Step 4: Formulate the Conclusion

Since the function satisfies the condition \(f(-x) = -f(x)\) for all values of \(x\) in the domain given by the table, we can conclude that the function is odd.

Key Concepts

even functions

In mathematics, even functions are defined by a specific symmetry in their graphs. This symmetry can be understood through the formula:

• For a function \( f(x) \), it is considered even if and only if \( f(-x) = f(x) \) for all \( x \) in the function’s domain.
• This means that the graph of an even function is symmetric with respect to the y-axis.

Imagine a curve mirrored perfectly on the left-hand side of the y-axis appearing the same on the right. That would represent an even function.
Particularly, polynomial functions like \( f(x) = x^2 \) or \( f(x) = x^4 + 2 \) are often examples of even functions. Every term in these polynomials has an even exponent.

Exploring tables or graphs, ensures us whether the y-value for any negative x is the same as for its positive counterpart, confirming such symmetry.

function properties

When analyzing functions, it's crucial to understand their properties. These characteristics help predict and describe the behavior of functions. Here are some key properties:

• **Domain and Range:** This tells us where the function exists and the set of possible output values.
• **Periodic:** A function may repeat its values in regular intervals, commonly seen in trigonometric functions.
• **Monotonicity:** A function can be strictly increasing, decreasing, or neither over a range.
• **Symmetry:** Functions can be classified as even or odd based on symmetry properties.

Understanding whether a function is symmetric (like even or odd) can significantly simplify calculations and provide insights into its graph.
Each function property acts as a clue guiding us to understand the full nature of a function.

precalculus

Precalculus is a foundational course that prepares students for the concepts they will encounter in calculus. It introduces a wide array of mathematical topics that establish the groundwork necessary for understanding calculus. Key areas of study include:

• **Functions:** Recognizing different types of functions and their properties, like even and odd functions, is a primary focus.
• **Sequences and Series:** Understanding arithmetic and geometric progressions, as well as limits, begins at this stage.
• **Trigonometry:** This involves studying angles and their relationships, essential for calculus.
• **Analytic Geometry:** This often covers the geometry of lines, circles, and conics.

Precalculus fosters a deeper comprehension by illuminating the principles underlying calculus.
The blend of algebraic and geometric perspectives sets the stage for tackling advanced calculus topics. A firm grasp of functions, including symmetry properties, enhances problem-solving skills and facilitates a smoother transition to calculus studies.