Problem 40 - Precalculus Essentials

Question

Determine whether each function is even, odd, or neither. Then determine whether the function's graph is symmetric with respect to the \(y\) -axis, the origin, or neither. $$g(x)=x^{2}-x$$

Step-by-Step Solution

Answer

The function \(g(x) = x^{2} - x\) is neither even nor odd, and its graph is not symmetric with respect to the y-axis or the origin.

Step 1: Apply the Even Function Test

An even function satisfies the property \(g(-x) = g(x)\). For the given function \(g(x) = x^{2} - x\), substitute \(-x\) for x and simplify the expression: \(g(-x) = (-x)^{2} - (-x) = x^{2} + x\). This is not equal to \(g(x)\), therefore the function is not even.

Step 2: Apply the Odd Function Test

An odd function satisfies the property \(g(-x) = -g(x)\). We already have \(g(-x) = x^{2} + x\). Now, compute \(-g(x) = - (x^{2} - x) = -x^{2} + x\). The result shows \(g(-x)\) is not equal to \(-g(x)\), therefore the function is neither odd.

Step 3: Evaluate the Graph Symmetry

As the function is neither even nor odd, the graph of the function \(g(x) = x^{2} - x\) is not symmetric with respect to the y-axis (which would have been the case if it were even) nor with respect to the origin (which would have been the case if it were odd). Thus, the graph of the function has no symmetry.