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Chapter 0: Problem 53

Determine whether the function is even, odd, or neither. $$ f(x)=\frac{|x|+1}{x^{4}-2 x^{2}+3} $$

Short Answer

Expert verified
The function \(f(x)=\frac{|x|+1}{x^{4}-2 x^{2}+3}\) is even since after substituting x with -x, the function remains the same: \(f(-x) = f(x)\).

Step by step solution

01

Replace x with -x

Replace each x in the given function with -x: \[ f(-x)=\frac{|-x|+1}{(-x)^{4}-2(-x)^{2}+3} \]
02

Simplify the function

Simplify the function obtained in Step 1: \[ f(-x)=\frac{|-x|+1}{x^{4}-2 x^{2}+3} \]
03

Compare the new function with the original function

We can see that the simplified function obtained after replacing x with -x is the same as the original function: \[ f(-x)=\frac{|-x|+1}{x^{4}-2 x^{2}+3} = f(x) \] Since f(-x) = f(x), the function is even.
04

Conclusion

The function \(f(x)=\frac{|x|+1}{x^{4}-2 x^{2}+3}\) is even.

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