Question

Determine whether is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\frac{{{{\bf{x}}^{\bf{2}}}}}{{{{\bf{x}}^{\bf{4}}}{\bf{ + 1}}}}\)

Short answer

The function \({\rm{f}}\left( {\rm{x}} \right){\rm{ = }}\left( {\frac{{{{\rm{x}}^{\rm{2}}}}}{{{{\rm{x}}^{\rm{4}}}{\rm{ + 1}}}}} \right)\) is an even function.

Step-by-step solution

Determination of even or odd for given function theoretically

The given function will even function if\({\bf{f}}\left( {{\bf{ - x}}} \right){\bf{ = f}}\left( {\bf{x}} \right)\). The given function will be odd if\({\bf{f}}\left( {{\bf{ - x}}} \right){\bf{ = - f}}\left( {\bf{x}} \right)\).

Replace x by –x for a given function to check for odd or even.

\(\begin{array}{c}f\left( { - x} \right) = \frac{{{{\left( { - x} \right)}^2}}}{{{{\left( { - x} \right)}^4} + 1}}\\f\left( { - x} \right) = \frac{{{x^2}}}{{{x^4} + 1}}\\f\left( { - x} \right) = \left( {\frac{{{x^2}}}{{{x^4} + 1}}} \right)\\f\left( { - x} \right) = f\left( x \right)\end{array}\)

Since\(f\left( { - x} \right) = f\left( x \right)\), therefore given function is an even function.

Verification of answer graphically

The graph of the given function is shown below:

Since the graph of the given function is symmetric about the y-axis. So the given function is even.

Therefore, the given function is an even function.