Worked Example Even, Odd, Neither Even Nor Odd Function

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Worked example Even, Odd, Neither even nor odd function

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<ul><li><p>Test to determine if a function is even, odd or neither </p><p>1 </p><p>Mathematics Section Universiti Kuala Lumpur Malaysia France Institute </p><p>How to determine if a function is even, odd or neither when the FUNCTION is given </p><p>algebraically, )x(fy </p><p>To test whether a function )x(fy is even, odd or neither even nor odd, replace x with x </p><p>and compare the result to )x(f </p><p> If )x(f)x(f then the function )x(f is even </p><p> If )x(f)x(f then the function )x(f is odd </p><p> If )x(f)x(f and )x(f)x(f then the function )x(f is neither even nor odd </p><p>Example 1 </p><p>Determine whether 7x3x)x(f 24 is even, odd or neither even nor odd function. </p><p>STEP 1: Replace x with -x </p><p> 7x3x)x(f 24 </p><p> 7)x(3)x()x(f 24 </p><p>STEP 2: Recalculate </p><p> 7x3x)x(f 24 </p><p>STEP 3: Compare the result to )x(f </p><p> )x(f 7x3x24 )x(f </p><p>STEP 4: Conclusion </p><p>Since )x(f)x(f , the function 7x3x)x(f 24 is </p><p>an EVEN function </p></li><li><p>Test to determine if a function is even, odd or neither </p><p>2 </p><p>Mathematics Section Universiti Kuala Lumpur Malaysia France Institute </p><p>Example 2 </p><p>Determine whether xx</p><p>4x)x(f</p><p>3</p><p>2</p><p> is even, odd or neither even nor odd function. </p><p>STEP 1: Replace x with -x </p><p>xx</p><p>4x)x(f</p><p>3</p><p>2</p><p> )x()x(</p><p>4)x()x(f</p><p>3</p><p>2</p><p>STEP 2: Recalculate </p><p> xx</p><p>4x)x(f</p><p>3</p><p>2</p><p>STEP 3: Compare the result to )x(f </p><p> xx</p><p>4x)x(f</p><p>3</p><p>2</p><p> )x(f </p><p>Conclusion: </p><p>Since )x(f)x(f , the function xx</p><p>4x)x(f</p><p>3</p><p>2</p><p> is NOT </p><p>AN EVEN function </p><p>STEP 4: Try to factor out 1 </p><p>xx</p><p>4x</p><p>)xx(</p><p>4x)x(f</p><p>3</p><p>2</p><p>3</p><p>2</p><p>STEP 5: Compare the result to )x(f </p><p>xx</p><p>4x)x(f</p><p>3</p><p>2</p><p>)x(f </p><p> STEP 6: Conclusion Since </p><p>)x(f)x(f , the function xx</p><p>4x)x(f</p><p>3</p><p>2</p><p> is an </p><p>ODD function </p></li><li><p>Test to determine if a function is even, odd or neither </p><p>3 </p><p>Mathematics Section Universiti Kuala Lumpur Malaysia France Institute </p><p>Example 3 </p><p>Determine whether 8x3x)x(f 35 is even, odd or neither even nor odd function. </p><p>STEP 1: Replace x with -x </p><p> 8x3x)x(f 35 </p><p> 8)x(3)x()x(f 35 </p><p>STEP 2: Recalculate </p><p> 8x3x)x(f 35 </p><p>STEP 3: Compare the result to )x(f </p><p> 8x3x)x(f 35 )x(f </p><p>Conclusion: </p><p>Since )x(f)x(f , the function 8x3x)x(f 35 is </p><p>NOT AN EVEN function </p><p>STEP 4: Try to factor out 1 </p><p> )8x3x(8x3x)x(f 3535 </p><p>STEP 5: Compare the result to )x(f </p><p> )8x3x()x(f 35 )x(f </p><p> Conclusion: </p><p>Since )x(f)x(f , the function 8x3x)x(f 35 is </p><p>NOT AN ODD function </p><p> STEP 6: Conclusion </p><p>Since )x(f)x(f and )x(f)x(f , the function </p><p>8x3x)x(f 35 is NEITHER even nor odd </p><p>function </p></li><li><p>Test to determine if a function is even, odd or neither </p><p>4 </p><p>Mathematics Section Universiti Kuala Lumpur Malaysia France Institute </p><p>How to determine if a function is even, odd or neither when the GRAPH of the </p><p>function is given </p><p> Any function with a graph symmetric about the y-axis is an EVEN function </p><p> Any function with a graph symmetric about the origin is an ODD function </p><p> Any function with a graph NON symmetric about the y-axis or the origin is a </p><p>NEITHER EVEN NOR ODD function </p><p>Example 4 </p><p>Based on the following graphs, determine whether the function is even, odd or neither even nor </p><p>odd function. </p><p> (a) </p><p> (b) </p><p> (c) </p><p> Since the graph is symmetric about the y-axis, the function is even. </p><p> Since the graph is symmetric about the origin, the function is odd. </p><p> Since the graph is NOT symmetric about the y-axis and the origin, the function is neither even nor odd. </p></li></ul>