Help me with this exercise please

Question

Determine symmetry with respect to the x y axis or origin, if any exists, and graph..

y = 1/2 x 1

Answer 1

Do you mean y = 1/2x + 1?

If f(-x) = f(x), then f(x) is even (symmetric about the y-axis).

If f(-x) = -f(x), then f(x) is odd (symmetric about the origin).

If y = f(x) and -y = f(x), then f(x) is symmetric about the x-axis.

We have f(x) = 1/2x + 1

Replace x with -x.

f(-x) = 1/2(-x) + 1

f(-x) = -1/2x + 1

This doesn't equal 1/2x+1 or -1/2x-1, so it is neither even or odd.

Also, if we have y = 1/2x + 1, and we replace y with -y, then

-y = 1/2x + 1

So y = -1/2x - 1 which is not what we started with.

So it is not symmetric about the x-axis.