To answer this question, we need to remember the following key concept:
• A function is even if we check that,:
[tex]f(-x)=f(x)[/tex]
In this case, we can say that the function has been reflected in the y-axis.
• Likewise, we can say that a function is odd if we check that:
[tex]f(-x)=-f(x)[/tex]
Then, we need to check these two situations with the three given functions as follows:
First case
[tex]f(x)=\frac{2}{3}x+1[/tex]
1. We need to check if the function is even:
[tex]f(-x)=\frac{2}{3}(-x)+1=-\frac{2}{3}x+1\Rightarrow f(-x)=-\frac{2}{3}x+1[/tex]
We can check that:
[tex]f(-x)\ne f(x)[/tex][tex]f(x)=\frac{2}{3}x+1\ne f(-x)=-\frac{2}{3}x+1[/tex]
Therefore, the function is not even.
2. We need to verify if the function is odd:
[tex]f(-x)=-f(x)[/tex][tex]-f(x)=-(\frac{2}{3}x+1)=-\frac{2}{3}x-1[/tex]
Then, we have:
[tex]f(-x)=-\frac{2}{3}x+1\ne-f(x)=-\frac{2}{3}x-1[/tex]
Then, the function is not odd.
Therefore, the function is neither even nor odd.
Second case
We can proceed in a similar way as before:
1. We need to verify if the function is even:
[tex]f(x)=f(-x)[/tex][tex]f(x)=(x+2)^2=x^2+4x+4[/tex][tex]f(-x)=(-x)^2+4(-x)+4=x^2-4x+4[/tex]
Then
[tex]f(x)\ne f(-x)[/tex]
Since
[tex]f(x)=x^2+4x+4\ne f(-x)=x^2-4x+4[/tex]
Thus, the function is not even.
2. Verify if the function is odd
[tex]f(-x)=-f(x)[/tex][tex]-f(x)=-(x^2+4x+4)=-x^2-4x-4[/tex]
Then, we have that:
[tex]f(-x)\ne-f(x)\Rightarrow f(-x)=x^2-4x+4\ne-x^2-4x-4[/tex]
Hence, the function is not odd.
Therefore, the function is neither even nor odd.
Third Case
We have the function:
[tex]f(x)=|x|-x^2[/tex]
We need to remember that |x| is the function absolute value.
1. Is the function even?
Then, we have:
[tex]f(-x)=|-x|-(-x)^2=|-(-x)|-x^2=|x|-x^2[/tex]
Then, we can see that the function is even, since:
[tex]f(x)=f(-x)\Rightarrow f(x)=|x|-x^2=f(-x)=|x|-x^2[/tex]
Then, the function is even.
2. Is the function odd?
[tex]-f(x)=-(|x|-x^2)=-|x|+x^2[/tex]
Then, we have:
[tex]f(-x)\ne-f(x)\Rightarrow f(-x)=|x|-x^2\ne-|x|+x^2[/tex]
Thus, the function is not odd.
Therefore, this function is even. However, it is not odd.
In summary, we have:
a. The function:
[tex]f(x)=\frac{2}{3}x+1[/tex]
Neither even nor odd.
b. The function:
[tex]f(x)=(x+2)^2[/tex]
Neither even nor odd.
c. The function:
[tex]f(x)=|x|-x^2[/tex]
The function is even. However, it is not odd.
