Question

O GRAPHS AND FUNCTIONSEven and odd functions: Problem type 1

Answer 1

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:

`f(-x)=-f(x)`

And we can say that the function is symmetric with respect to the origin.

• A function is even if we have that:

`f(x)=f(-x)`

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:

`g(x)=5x^2`

Then to determine if it is even we have:

`\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}`

Therefore, this function is even.

We can also determine if the function is odd by using a similar procedure:

`\begin{gathered} g(-x)=-g(x) \\ \text{ We already got that }g(-x)=g(x)\\e-g(x) \\ \end{gathered}`

Therefore, the function is NOT an odd function.

Function h(x)

To determine if the function is even, we have:

`\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)\\e h(-x) \end{gathered}`

Then the function is NOT even.

Now, we have to determine if the function is odd:

`\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)\\e h(-x) \\ \end{gathered}`

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