Is the following function symmetric with respect to the y-axis, symmetric with respect to the x-axis, symmetric with respect to the origin, or none of the above?
f(x)=-5x^3+2x
I think it is:
-3+1=2
This function is symmetric to the y-axis.
If it were symmetric with respect to the y axis, then at any x, for example x = 1, y would be the same for + x and for -x
So let's try x = 1
y = -5 + 2 = -3
now for x = -1
y = +5 -2 = +3
Nope, sorry
To determine the symmetry of a function, you need to consider how the function behaves when certain transformations are applied to it.
To check for symmetry with respect to the y-axis, you compare the function values for positive and negative x-values. In other words, if substituting x for -x in the function results in the same function value, then the function is symmetric with respect to the y-axis.
Here's how you can determine the symmetry of the function f(x) = -5x^3 + 2x using the y-axis test:
1. Substitute -x for x in the given function:
f(-x) = -5(-x)^3 + 2(-x)
2. Simplify the function:
f(-x) = -5(-x)^3 - 2x
= -5(-x)(-x)(-x) - 2x
= -(-5x^3) - 2x
= 5x^3 - 2x
3. Compare the simplified function with the original function:
f(x) = -5x^3 + 2x
f(-x) = 5x^3 - 2x
The function f(x) = -5x^3 + 2x is not equivalent to f(-x) = 5x^3 - 2x, which means the function does not satisfy the condition for symmetry with respect to the y-axis.
Hence, your initial assessment that the function is symmetric to the y-axis is incorrect.