Given $y = \frac { x ^ { 3 } } { x ^ { 4 } + 1 }$ , use the algebraic tests to determine symmetry with respect to both axes and the origin.
A)y-axis symmetry only
B)x-axis symmetry only
C)origin symmetry only
D)x-axis, y-axis, and origin symmetry
E)no symmetry

Question

Quizplus · Accepted Answer

To determine symmetry, we use algebraic tests. For y-axis symmetry, we replace $x$ with $-x$ and see if we get the original equation. For x-axis symmetry, we replace $y$ with $-y$ and see if we get the original equation. For origin symmetry, we replace both $x$ with $-x$ and $y$ with $-y$ and see if we get the original equation.1. **Y-axis symmetry**: Replace $x$ with $-x$:$$y = \frac{(-x)^3}{(-x)^4 + 1} = \frac{-x^3}{x^4 + 1}$$This is not the same as the original equation, so there's no y-axis symmetry.2. **X-axis symmetry**: Replace $y$ with $-y$:$$-y = \frac{x^3}{x^4 + 1}$$This is not the same as the original equation, so there's no x-axis symmetry.3. **Origin symmetry**: Replace $x$ with $-x$ and $y$ with $-y$:$$-y = \frac{(-x)^3}{(-x)^4 + 1} = \frac{-x^3}{x^4 + 1}$$This is the same as replacing $y$ with $-y$ in the original equation, indicating origin symmetry.Therefore, the function has origin symmetry only.

Given y=x3x4+1y = \frac { x ^ { 3 } } { x ^ { 4 } + 1 }y=x4+1x3​

Given $y = \frac { x ^ { 3 } } { x ^ { 4 } + 1 }$