98 points possible Use the algebraic tests to check for symmetry with respect to both axes and the origin. y = (x)/(x^(2)+4 )

Question

98 points possible

Use the algebraic tests to check for symmetry with respect to both axes and the origin.
y =
`(x)/(x^(2)+4 )`

Answer 1

Explanation:

symmetry with respect to y-axis for y=f(x) means f(-x)=f(x)

in this case, y = f(x) = x / (x^2+4)

f(-x) = -x / ((-x)^2+4) = -x / (x^2+4) = -f(x)

so it is not symmetric to y-axis

symmetry with respect to x-axis for x=g(y) means g(-y)=g(y)

in this case, y = x / (x^2+4)

y*(x^2+4) = x

y*x^2 + 4y - x = 0

substitute -y into g(y)

(-y)*x^2 +4(-y) - x = 0

-y*x^2 - 4y - x = 0

y*x^2 + 4y + x = 0

so g(-y) is not equal to g(y)

so it is not symmetric to x-axis

Answer 2

Explanation:

I will test it for symmetry with respect to the origin; which means for y=f(x)

f(-x) = -f(x)

f(-x) = -x / ((-x)^2 + 4)

= -x / (x^2 + 4)

= -f(x)

So it is proven that the expression is symmetric to the origin.