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To determine whether the function \( f(x) = 5\cos(x) - 4 \) is even or odd, we need to apply the following tests:

1. **Even function**: A function is even if \( f(-x) = f(x) \).
2. **Odd function**: A function is odd if \( f(-x) = -f(x) \).

Let's calculate \( f(-x) \):

\[
f(-x) = 5 \cos(-x) - 4
\]

Since \( \cos(-x) = \cos(x) \) (because cosine is an even function), we get:

\[
f(-x) = 5 \cos(x) - 4
\]

This is exactly the same as \( f(x) \), so we conclude:

\[
f(-x) = f(x)
\]

Thus, the function \( f(x) = 5 \cos(x) - 4 \) is an **even function**.

Would you like further details or have any questions? 

Here are 5 related questions to expand this concept:

1. How would the analysis change if we had a sine function instead of cosine?
2. Can a function be both even and odd? Why or why not?
3. How does the vertical shift (the \(-4\)) affect whether the function is even or odd?
4. What other transformations affect the even or odd nature of a function?
5. How would the graph of an even function like this one look in terms of symmetry?

**Tip**: Even functions are symmetric with respect to the y-axis, while odd functions are symmetric about the origin.

Determine whether the function f(x) = 5 cos(x) - 4 is even or odd by applying the rules for function symmetry. This analysis involves core concepts from trigonometry, specifically related to cosine functions and their properties. Suitable for Grades 10-12.

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