list the intercepts and test for symmetry y = x^4 + 1 divided by 2x^5
To find the x-intercepts of a function, we need to set y = 0 and solve for x.
For the given equation, y = (x^4 + 1) / (2x^5), let's set y = 0 and solve for x:
0 = (x^4 + 1) / (2x^5)
Multiplying both sides of the equation by 2x^5 to eliminate the denominator:
0 * (2x^5) = (x^4 + 1)
0 = x^4 + 1
Subtracting 1 from both sides:
-1 = x^4
Taking the fourth root of both sides:
±√(-1) = x
Since we cannot take square root of negative numbers in real numbers, there are no real x-intercepts. The function has no points where it crosses the x-axis.
To find the y-intercept, we set x = 0 and solve for y:
y = (0^4 + 1) / (2 * 0^5)
y = 1 / 0
Dividing by zero is undefined, which means that there is no y-intercept for this function.
Now, let's test for symmetry.
To test for symmetry, we can examine the function for symmetry with respect to the y-axis (even symmetry) and the x-axis (odd symmetry).
1. Even symmetry (symmetry with respect to the y-axis):
We replace x with -x in the equation and simplify. If the equation remains the same, the function has even symmetry.
Replacing x with -x in the equation y = (x^4 + 1) / (2x^5), we get:
y = (-x)^4 + 1 / 2(-x)^5
y = x^4 + 1 / 2(-x)^5
Since the equation does not remain the same, the function does not have even symmetry.
2. Odd symmetry (symmetry with respect to the x-axis):
We replace y with -y in the equation and simplify. If the equation remains the same, the function has odd symmetry.
Replacing y with -y in the equation y = (x^4 + 1) / (2x^5), we get:
-y = (x^4 + 1) / (2x^5)
Multiplying both sides by -1 to remove the negative sign:
y = -(x^4 + 1) / (2x^5)
Since the equation does not remain the same, the function does not have odd symmetry either.
In conclusion, the given function, y = (x^4 + 1) / (2x^5), has no x-intercepts, no y-intercept, and does not exhibit even or odd symmetry.