Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=\frac{x^{2}+5}{x}$$

Step-by-Step Solution

Answer

The function is odd.

Step 1: Substitute \(-x\) into the function

To analyze whether the function is even or odd, we first evaluate \(f(-x)\) by substituting \(-x\) into the function. This gives us: \[ f(-x) = \frac{(-x)^2 + 5}{-x} \] Simplifying this, we find, \[ f(-x) = \frac{x^2 + 5}{-x} = -\frac{x^2 + 5}{x} \]

Step 2: Compare \(f(-x)\) to \(f(x)\)

Now, let's compare the modified function \(f(-x) = -\frac{x^2 + 5}{x}\) with the original function \(f(x) = \frac{x^2 + 5}{x}\). An odd function satisfies \(f(-x) = -f(x)\), while an even function satisfies \(f(-x) = f(x)\).

Step 3: Determine the Function Type

Notice that \(f(-x) = -\frac{x^2 + 5}{x}\) is equal to \(-\left(\frac{x^2 + 5}{x}\right) = -f(x)\). This satisfies the condition for odd functions, where \(f(-x) = -f(x)\). Thus, the function is odd.

Key Concepts

Even Functions

In mathematics, an even function is a function that remains unchanged if you substitute the variable with its negative counterpart. This means if you have a function \( f(x) \), then \( f(x) \) is even if \( f(-x) = f(x) \) for all values of \( x \). Such functions exhibit symmetry about the y-axis.
Common examples of even functions include:

• The quadratic function \( f(x) = x^2 \)
• The cosine function \( f(x) = \cos(x) \)

Even functions are called so because they are symmetric; if you were to graph them, it would look the same on the left side of the y-axis as it does on the right. For the function given in the exercise, evaluating \( f(-x) \) leads us to conclude that it is not an even function, because \( f(-x) eq f(x) \).
This understanding helps us classify functions and predict their behavior, which is useful in calculus, engineering, and other fields.

Odd Functions

Odd functions have a special characteristic known as point symmetry about the origin. For a function \( f(x) \) to be considered odd, \( f(-x) \) must equal \(-f(x) \) for all values of \( x \). This means that if you take a point to the right of the y-axis, its corresponding point to the left of the y-axis will reflect through the origin (0,0) on a graph.
Some well-known examples of odd functions are:

• The cubic function \( f(x) = x^3 \)
• The sine function \( f(x) = \sin(x) \)

In the original exercise with \( f(x) = \frac{x^2+5}{x} \), we determined it was an odd function by evaluating \( f(-x) \). The result clearly established that \( f(-x) = -f(x) \). This thorough analysis shows the value of understanding function types in predicting how they behave across different mathematical problems.

Function Evaluation

Function evaluation is an essential concept in mathematics as it involves plugging in values or expressions into a function to obtain results. This method helps in determining if a function satisfies particular properties, such as being even or odd.
To evaluate a function like \( f(x) = \frac{x^2+5}{x} \), you may need to substitute values, such as \( -x \) for \( x \), to explore the behavior of the function under different circumstances.
Steps in function evaluation include:

• Identify the function and its variable.
• Substitute the value or expression into the function.
• Simplify the expression to see if it matches any known conditions (like \( f(-x) = f(x) \) or \( f(-x) = -f(x) \)).

In the context of the given exercise, by evaluating \( f(-x) \), we gained insight into the odd nature of the function. Such analysis is crucial in fields like calculus and algebra, where evaluating functions lets us analyze and understand the properties and behavior of equations and formulas.