Even Function

An even function is a mathematical function that satisfies the property f(x) = f(-x) for all values of x in its domain. In other words, if you replace x with -x in the function, the result remains unchanged. Geometrically, even functions are symmetric with respect to the y-axis.

Analysis of Options

Let's analyze each option to determine which one is an even function:

a) x^8
- If we replace x with -x in the function, we get (-x)^8 = x^8.
- The function remains unchanged when x is replaced with -x, so it satisfies the condition f(x) = f(-x).
- Therefore, option 'A' is an even function.

b) x^3
- If we replace x with -x in the function, we get (-x)^3 = -x^3.
- The function changes sign when x is replaced with -x, so it does not satisfy the condition f(x) = f(-x).
- Therefore, option 'B' is not an even function.

c) x^33
- If we replace x with -x in the function, we get (-x)^33 = -x^33.
- The function changes sign when x is replaced with -x, so it does not satisfy the condition f(x) = f(-x).
- Therefore, option 'C' is not an even function.

d) x^73
- If we replace x with -x in the function, we get (-x)^73 = -x^73.
- The function changes sign when x is replaced with -x, so it does not satisfy the condition f(x) = f(-x).
- Therefore, option 'D' is not an even function.

Conclusion

Among the given options, only option 'A' (x^8) is an even function as it satisfies the condition f(x) = f(-x). The other options do not exhibit this property.