Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{x^{2}+3}{|x|}$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Evaluate f(-x)

Substitute (-x) for (x) in the function. You get: \[ f(-x) = \frac{(-x)^{2} + 3}{|-x|} \] After simplifying, (-x)^{2} = x^{2} and |−x| = |x|. So, \[ f(-x) = \frac{x^{2} + 3}{|x|} \]

Step 2: Compare f(-x) with f(x)

The expression for f(-x) is \( \frac{x^{2} + 3}{|x|} \) and this is the same expression as f(x). This indicates that f(-x) = f(x), suggesting that the function might be even.

Step 3: Determine the nature of the function

Since f(-x) = f(x) for all values of x where x is not equal to zero, the function f(x) is considered an even function. Therefore, it maintains symmetry about the y-axis.

Key Concepts

Function Symmetry

In mathematics, the concept of function symmetry is essential in understanding how a function behaves graphically. Symmetry can help predict values and simplify graphing. Two primary types of symmetries related to functions are even symmetry and odd symmetry.

When we say a function is even, we refer to its symmetry about the y-axis. An even function, defined by the condition \( f(-x) = f(x) \) for all \( x \) in the function's domain, will mirror itself on either side of the y-axis. This means that if you consider any point \( (x, y) \) on the graph of the function, the point \( (-x, y) \) will also lie on the graph.

• Example: The function \( f(x) = x^2 \) is even because \( f(-x) = (-x)^2 = x^2 = f(x) \).
• The function in the original exercise, \( f(x) = \frac{x^2 + 3}{|x|} \), is even as it satisfies the criterion \( f(-x) = f(x) \).

Identifying symmetry is crucial for understanding a function's behavior without necessarily plotting every point, which is particularly helpful in calculus and precalculus studies.

Function Evaluation

Function evaluation involves determining the outcome of a function given a specific input. Understanding how to evaluate functions is a fundamental skill in precalculus and other areas of mathematics.

To evaluate a function, you substitute the input variable \( x \) with a given value or parameter, such as \(-x \) in the original exercise. For instance, if given \( f(x) = x^2 + 2x + 1 \) and asked to find \( f(-2) \), you replace every \( x \) with \(-2\), leading to \((-2)^2 + 2(-2) + 1\).

• This results in \( f(-2) = 4 - 4 + 1 = 1 \).

In the exercise, evaluating \( f(-x) \) involved substituting \(-x\) into the function \( f(x) = \frac{x^2 + 3}{|x|} \) and simplifying the expression. This step is vital to perform correctly to determine the function's properties, such as identifying symmetry.
By understanding function evaluation, you can explore how changing inputs affect the output, analyze function properties, and delve into functional analysis.

Precalculus Concepts

Precalculus serves as a bridge between algebra and calculus, encompassing various essential concepts that prepare students for more advanced mathematical studies. Some key areas include functions, trigonometry, sequences, and limits.

One critical aspect of precalculus is mastery of different types of functions and their properties. This includes understanding linear, quadratic, polynomial, rational, and other function types. For instance, recognizing even and odd functions, as discussed in the original exercise, is a crucial precalculus concept.

• Understanding the graphical representation of functions, including recognizing symmetries and transformations, forms the basis for calculus.
• Function behavior, such as end behavior and intercepts, helps in sketching graphs accurately without relying solely on plotting points.

The exploration of functions in precalculus also introduces students to analytical techniques, such as factoring or synthetic division for algebraic expressions. Building this foundation is vital for tackling the limits, derivatives, and integrals of calculus, all of which rely heavily on a solid understanding of precalculus concepts.
To sum up, precalculus topics like function evaluation and symmetry play significant roles in equipping students with the tools to navigate through higher-level mathematics.