To determine whether a function is even, odd, or neither, we can apply the following tests:

1. Even function: A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$.
2. Odd function: A function $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$.

Given the function: $f(x) = x^2 - 1$

Step 1: Test for Even Function

Substitute $-x$ into the function: $f(-x) = (-x)^2 - 1 = x^2 - 1$

Since $f(-x) = f(x)$, the function satisfies the condition for being even.

Step 2: Test for Odd Function

Now, check if the function is odd by seeing if $f(-x) = -f(x)$: $f(-x) = x^2 - 1$ $-f(x) = -(x^2 - 1) = -x^2 + 1$

Since $f(-x) \neq -f(x)$, the function is not odd.

Conclusion:

The function $f(x) = x^2 - 1$ is even.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:

1. How do we test for odd and even functions with more complex expressions?
2. What types of graphs do even and odd functions typically have?
3. Can a function be both even and odd?
4. How would transformations (like shifting the graph) affect whether a function is even or odd?
5. What is the significance of even and odd functions in calculus?

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