Problem 64 - A Graphical Approach to Precalculus with Limits

Question

Each function is either even or odd Evaluate \(f(-x)\) to determine which situation applies. $$f(x)=8$$

Step-by-Step Solution

Answer

The function is even.

Step 1: Understand the Problem

We need to determine if the function \( f(x) = 8 \) is even or odd. To do this, we'll evaluate \( f(-x) \). A function is even if \( f(-x) = f(x) \) and odd if \( f(-x) = -f(x) \).

Step 2: Evaluate f(-x)

Since \( f(x) = 8 \), it means that for any input \( x \), the output is always 8. Thus, \( f(-x) = 8 \) as well, because \( 8 \) is a constant and does not depend on \( x \).

Step 3: Compare f(-x) to f(x)

We need to see if \( f(-x) = f(x) \) or \( f(-x) = -f(x) \). Since \( f(-x) = 8 \) and \( f(x) = 8 \), we have \( f(-x) = f(x) \).

Step 4: Determine if the Function is Even or Odd

The condition \( f(-x) = f(x) \) holds true. According to the definitions, this means the function is even.

Key Concepts

Evaluating Functions

Evaluating functions is like performing a test to see how the function reacts when we alter its input. When you have a function, usually expressed as \( f(x) \), and you plug in a specific value or change the variable to \(-x\), you're evaluating that function. This step is essential because it helps determine the behavior of the function.

For example, imagine you have a function \( f(x) = 8 \). To evaluate \( f(-x) \), you replace \( x \) with \(-x\). Here, since \( f(x) = 8 \), no matter what \( x \) is, \( f(-x) = 8 \) as well. Evaluating functions in such cases shows that the function's output isn't affected by the inputs, helping us deduce more about its nature.

Constant Function

A constant function is a unique type of function where the output value remains the same regardless of the input. In mathematical terms, if you have \( f(x) = c \), \( c \) being a number, then for any value of \( x \), \( f(x) \) always equals \( c \).

For instance, with \( f(x) = 8 \), whether you plug in \( x \), \(-x\), or any other number in the universe, the output will consistently be 8. Some key attributes of constant functions include:

• No matter the x-value, the function value is always constant.
• The graph of a constant function is a horizontal line across the y-axis at \( y = c \).

Constant functions are incredibly predictable, providing a solid foundation in understanding more complex functional relationships.

Function Symmetry

Understanding symmetry in functions helps identify whether a function is even or odd. Consider it like a mirror test—where you examine if the function reflects perfectly when you adjust the input by \(-x\).

An even function has symmetry around the y-axis. When you calculate \( f(-x) \) and find it equals \( f(x) \), the function is even. Suppose you have \( f(x) = 8 \). Here, \( f(-x) = 8 \) and \( f(x) = 8 \), both outputs are the same, confirming that it's even.

An odd function, in contrast, shows symmetry respectively to the origin. If \( f(-x) = -f(x) \), the function is odd. For the example \( f(x) = 8 \), this condition does not apply as \( -8 eq 8 \). Recognizing these symmetries is an economical way to analyze functions, making the complexities more accessible.