Problem 76 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=9$$

Step-by-Step Solution

Answer

The function is even because \(f(-x) = f(x) = 9\).

Step 1: Understand the Definitions

A function is even if for all values of \(x\), \(f(-x) = f(x)\). A function is odd if \(f(-x) = -f(x)\). We need to determine which case applies to \(f(x) = 9\).

Step 2: Substitute \(-x\) into the Function

Substitute \(-x\) into the given function: \(f(-x) = 9\).

Step 3: Compare with the Original Function

Compare \(f(-x) = 9\) with \(f(x) = 9\). Since they are equal, \(f(x) = 9\) is an even function, as it satisfies the condition \(f(-x) = f(x)\).

Step 4: Evaluate for Odd Function Criteria

Check if \(f(-x) = -f(x)\). For \(f(-x) = 9\) and \(-f(x) = -9\), they are not equal. Therefore, the function is not odd.

Key Concepts

Function Evaluation

Function evaluation is an essential tool in mathematics. It involves substituting a given value into a function to find the corresponding output. To evaluate the function, you simply replace the variable (usually denoted as \(x\)) with the specific value you are interested in. In our problem, we are given a constant function, \(f(x) = 9\), and need to evaluate \(f(-x)\). Since the function is constant, no matter what value \(x\) takes, the output will remain 9. Thus, \(f(-x) = 9\). This process confirms that evaluating functions doesn’t always involve complex calculations. With a constant function, the output remains the same for any input value, demonstrating simplicity and ease.

Function Properties

Understanding the properties of functions is crucial for classifying them as even or odd. These properties help determine how the function behaves when its input changes.

• **Even Functions**: Characterized by the symmetry about the y-axis, an even function satisfies \(f(-x) = f(x)\). This means that the function's graph looks the same on the left and right of the y-axis.
• **Odd Functions**: Defined by rotational symmetry about the origin, an odd function fulfills the condition \(f(-x) = -f(x)\). The graph of an odd function looks the same when rotated 180 degrees around the origin.

For the function \(f(x) = 9\), since substituting \(-x\) yields the same result as \(f(x)\), \(f(-x) = f(x) = 9\), it falls under even functions. This highlights that the properties of even functions are about invariance under input transformation.

Precalculus Analysis

Precalculus is the study of basic algebra and trigonometry concepts aimed at getting students ready for calculus. Within precalculus, analyzing functions is a key skill. In our example, analyzing the function \(f(x) = 9\) encompassed understanding what type of function it is. The function did not depend on the specific variable for outputs, making analysis straightforward due to its constancy.When delving into precalculus, being able to recognize and evaluate even and odd functions can greatly aid in sketching graphs and modeling real-world scenarios with functions. Overall, simple functions such as \(f(x) = 9\) can serve as an introduction to deeper exploration of how different inputs influence a function’s outputs, letting students develop intuition about more complex forms of analysis.