We have the following functions, given algebraically, and as a graph:
And we have to determine if the functions are even, odd, or neither.
To determine each case, we need to recall when a function is even, or odd as follows:
• A function is odd if we have that:
[tex]f(-x)=-f(x)[/tex]
And we can say that the function is symmetric with respect to the origin.
• A function is even if we have that:
[tex]f(x)=f(-x)[/tex]
And we can say that the function is symmetric with respect to the y-axis.
Then we can conclude from the graphs that (functions r and s):
Function r
For function r, the function r graphically is not symmetric with respect to the y-axis, and neither with respect to the origin. Therefore, the function is neither odd nor even function.
Function s
The function s is symmetric with respect to the origin, that is, the function looks in the same way right side up or upside down. Then the function s is an odd function.
Function g(x)
We can analyze this function algebraically as follows:
[tex]g(x)=5x^2[/tex]
Then to determine if it is even we have:
[tex]\begin{gathered} g(x)=g(-x) \\ \\ g(-x)=5(-x)^2=5(-1x)^2=5(-1)^2(x)^2=5(1)x^2=5x^2=g(x) \\ \\ \therefore g(-x)=g(x) \end{gathered}[/tex]
Therefore, this function is even.
We can also determine if the function is odd by using a similar procedure:
[tex]\begin{gathered} g(-x)=-g(x) \\ \text{ We already got that }g(-x)=g(x)\ne-g(x) \\ \end{gathered}[/tex]
Therefore, the function is NOT an odd function.
Function h(x)
To determine if the function is even, we have:
[tex]\begin{gathered} h(x)=h(-x) \\ \\ h(-x)=7(-x)^4-2(-x)^3=7(-1x)^4-2(-1x)^3 \\ \\ h(-x)=7(-1)^4x^4-2(-1)^3x^3=7(1)x^4-2(-1)x^3 \\ \\ h(-x)=7x^4+2x^3 \\ \\ \therefore h(x)\ne h(-x) \end{gathered}[/tex]
Then the function is NOT even.
Now, we have to determine if the function is odd:
[tex]\begin{gathered} h(-x)=-h(x) \\ \\ -h(x)=-(7x^4-2x^3)=2x^3-7x^4 \\ \\ \text{ From the previous result, we have that:} \\ \\ -h(x)\ne h(-x) \\ \end{gathered}[/tex]
Then the function is NOT odd.
Therefore, in summary, we can conclude that:
• Function r ---> Neither
,
• Function s ---> Odd
,
• Function g(x) ---> Even
,
• Function h(x) ---> Neither
