How do you determine if #F(x) = x + 3# is an even or odd function?

Question

Ariana Alvarez · Accepted Answer

neither even nor odd.

Check if F(x) is even or odd by taking into account the following. • F(x) is even if it equals F(-x). There is symmetry in functions about the y-axis. • F(x) is odd if F(-x) = - F(x). About the origin, odd functions exhibit half-turn symmetry. Check for even F(-x) is equal to (-x) + 3 (-x + 3 ≠ F(x) F(x) is not even since F(x) ≠ F(-x) Check for odd #-F(x)=-(x+3)=-x-3≠F(-x)# F(x) is not odd since F(-x) ≠ - F(x) F(x) is therefore neither odd nor even. graph{x+3 [-10, 10, -5, 5]}

Elijah Allen · Answer

undefined To determine if $ F(x) = x + 3 $ is an even or odd function, you can use the properties of even and odd functions.

1. **Even Function**: A function $ f(x) $ is even if $ f(-x) = f(x) $ for all $ x $ in the domain of $ f $.

2. **Odd Function**: A function $ f(x) $ is odd if $ f(-x) = -f(x) $ for all $ x $ in the domain of $ f $.

Now, let's apply these definitions to $ F(x) = x + 3 $:

1. Check for evenness:
   $$ F(-x) = -x + 3 $$
   $$ F(x) = x + 3 $$

Since $ F(-x) $ is not equal to $ F(x) $, $ F(x) = x + 3 $ is not an even function.

2. Check for oddness:
   $$ F(-x) = -x + 3 $$
   $$ -F(x) = -(x + 3) = -x - 3 $$

Since $ F(-x) $ is not equal to $ -F(x) $, $ F(x) = x + 3 $ is not an odd function.

Therefore, $ F(x) = x + 3 $ is neither even nor odd.