Question

What makes a function an even function? On a graph, write down an example of an even function.

Answer 1

A function \( f(x) \) is considered an even function if it satisfies the condition:

\[
f(-x) = f(x) \quad \text{for all } x \text{ in the domain of } f.
\]

This means that the function is symmetric with respect to the y-axis. In practical terms, if you were to fold the graph along the y-axis, both halves would match perfectly.

### Example of an Even Function

A classic example of an even function is the quadratic function:

\[
f(x) = x^2.
\]

### Graph of \( f(x) = x^2 \)

On a graph, the function \( f(x) = x^2 \) looks like this:

- For positive values of \( x \): \( f(1) = 1^2 = 1 \)
- For negative values of \( x \): \( f(-1) = (-1)^2 = 1 \)

This shows that \( f(1) = f(-1) \).

Here's a representation of the graph:

```
| *
| *
| *
| *
------- | -----------------
| *
| *
| *
```

As you can see, both sides of the y-axis are mirror images, confirming that \( f(x) = x^2 \) is indeed an even function.