Respuesta :

konrad509

1.

[tex]f(x)=\sqrt{x^2}-9=|x|-9\\f(-x)=|-x|-9=|x|-9\\f(x)=f(-x)\implies\text{even}[/tex]

2.

[tex]g(x)=|x-3|\\g(-x)=|-x-3|\\g(x)\not=g(-x)\implies\text{not even}\\\\-g(x)=-|x-3|\\g(-x)\not=-g(x)\implies\text{not odd}[/tex]

3.

[tex]f(x)=\dfrac{x}{x^2-1}\\\\f(-x)=\dfrac{-x}{(-x)^2-1}=-\dfrac{x}{x^2-1}\\\\f(x)\not =f(-x)\implies \text{not even}\\\\-f(x)=-\dfrac{x}{x^2-1}\\f(-x)=-f(x)\implies \text{odd}[/tex]

4.

[tex]g(x)=x+x^2\\g(-x)=-x+(-x)^2=-x+x^2\\g(x)\not =g(-x)\implies \text{not even}\\\\-g(x)=-(x+x^2)=-x-x^2\\g(-x)\not =-g(x)\implies \text{not odd}[/tex]

Answer:

even

neither

odd

neither

Step-by-step explanation:

f(-x)=f(x) means f is even

f(-x)=-f(x) means f is odd

[tex]f(x)=\sqrt{x^2}-9[/tex]

Plug in -x.

[tex]f(-x)=\sqrt{(-x)^2}-9[/tex]

(-x)^2=x^2 so we have

[tex]f(-x)=\sqrt{x^2}-9[/tex]

This was the same function we started with f(x), so this function is even.

[tex]g(x)=|x-3|[/tex]

Plug in -x

[tex]g(-x)=|-x-3|[/tex]

[tex]g(-x)=|(-1)(x+3)|[/tex]

[tex]g(-x)=|(-1)||x+3|[/tex]

[tex]g(-x)=|x+3|[/tex]

We did not get the opposite or the same function back, so this is neither.

The opposite would have looked like this -|x-3|.

[tex]f(x)=\frac{x}{x^2-1}[/tex]

Plug in -x.

[tex]f(-x)=\frac{-x}{(-x)^2-1}[/tex]

[tex]f(-x)=\frac{-x}{x^2-1}[/tex] since (-x)^2 is the same as x^2

[tex]f(-x)=- \frac{x}{x^2-1}[/tex]

We got the opposite back so it is odd.

[tex]g(x)=x+x^2[/tex]

Plug in -x.

[tex]g(-x)=-x+(-x)^2[/tex]

(-x)^2=x^2

[tex]g(-x)=-x+x^2[/tex]

[tex]g(-x)=-(x-x^2)[/tex]

So we neither got the same or opposite back of the function, so it is neither.  The opposite would have looked like this -(x+x^2) or -x-x^2

So

even

neither

odd

neither

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