Question

1. error analysis describe and correct the error in determining whether the function $f(x) = x^2 + 3$ is even, odd, or neither. $f(x)=x^2 + 3$ $f(-x)=(-x)^2 + 3$ $=x^2 + 3$ $=f(x)$ so, $f(x)$ is an odd function.

Explanation:

Step1: Recall definitions of even/odd functions

A function \( f(x) \) is even if \( f(-x)=f(x) \) for all \( x \) in the domain. A function \( f(x) \) is odd if \( f(-x)=-f(x) \) for all \( x \) in the domain.

Step2: Analyze the given solution's error

The calculation of \( f(-x) \) is correct (\( f(-x)=(-x)^2 + 3=x^2 + 3=f(x) \)), but the conclusion is wrong. Since \( f(-x)=f(x) \), the function should be even, not odd.

Step3: Correct the conclusion

Based on the definition of an even function (since \( f(-x) = f(x) \)), \( f(x)=x^2 + 3 \) is an even function.

Answer:

The error is in the conclusion: since \( f(-x)=f(x) \), \( f(x) \) is even (not odd). The correct classification is that \( f(x)=x^2 + 3 \) is an even function.