Determine whether the following functions are even, odd, or neither algebraically.

Steps to Solve

1. Evaluate ( f(-x) ) for each function
   First, you'll need to substitute (-x) in place of (x) in each function and simplify.

2. Check the definitions of even and odd

• Even: A function is even if ( f(-x) = f(x) ) for all ( x ).
• Odd: A function is odd if ( f(-x) = -f(x) ) for all ( x ).
• Neither: If it doesn't fit either condition, it's neither.

3. Function 1: ( f(x) = x^2 + 2x + 1 )

   • ( f(-x) = (-x)^2 + 2(-x) + 1 = x^2 - 2x + 1 )
   • Compare: ( f(-x) \neq f(x) ) and ( f(-x) \neq -f(x) )

4. Function 2: ( f(x) = 7x^3 - x )

   • ( f(-x) = 7(-x)^3 - (-x) = -7x^3 + x )
   • Compare: ( f(-x) = -f(x) )

5. Function 3: ( f(x) = x^5 + 1 )

   • ( f(-x) = (-x)^5 + 1 = -x^5 + 1 )
   • Compare: ( f(-x) \neq f(x) ) and ( f(-x) \neq -f(x) )

6. Function 4: ( f(x) = x \sqrt{4 - x^2} )

   • ( f(-x) = -x \sqrt{4 - (-x)^2} = -x \sqrt{4 - x^2} )
   • Compare: ( f(-x) = -f(x) )

7. Function 5: ( f(x) = x^4 \sqrt{1 - x} + x )

   • ( f(-x) = (-x)^4 \sqrt{1 - (-x)} + (-x) = x^4 \sqrt{1 + x} - x )
   • Compare: ( f(-x) \neq f(x) ) and ( f(-x) \neq -f(x) )

8. Function 6: ( f(x) = |x| - 1 )

   • ( f(-x) = |-x| - 1 = |x| - 1 )
   • Compare: ( f(-x) = f(x) )

9. Function 7: ( f(x) = \frac{1}{4}x^6 - 5x^2 )

   • ( f(-x) = \frac{1}{4}(-x)^6 - 5(-x)^2 = \frac{1}{4}x^6 - 5x^2 )
   • Compare: ( f(-x) = f(x) )