Question

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

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

Short answer

The given function is neither odd nor even.

Step-by-step solution

Determination of even or odd for the given function

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.

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

Here,\(f\left( { - x} \right) \ne f\left( x \right)\)or\(f\left( { - x} \right) \ne - f\left( x \right)\)

Therefore, the given function is neither odd nor even.

Verification of answer graphically

The graph of the given odd is shown below:

Since the graph of the given function is neither symmetric about origin nor symmetric about the y-axis. So the given function is neither odd nor even.

Therefore, the given function is neither odd nor even.