Question

List the intercepts and test for symmetry. $$ y=\sqrt[5]{x} $$

Short answer

Intercepts: (0,0). Symmetric about the origin.

Step-by-step solution

- Find the x-intercept

To find the x-intercept, we set y = 0 and solve for x. The equation is \( y = \sqrt[5]{x} \). Setting y to 0, we get \( 0 = \sqrt[5]{x} \), which implies that \( x = 0 \). Therefore, the x-intercept is (0, 0).

- Find the y-intercept

To find the y-intercept, we set x = 0 and solve for y. The equation is \( y = \sqrt[5]{x} \). Setting x to 0, we get \( y = \sqrt[5]{0} = 0 \). Therefore, the y-intercept is (0, 0).

- Test for symmetry about the y-axis

To test for symmetry about the y-axis, we replace x with -x in the equation and see if the equation remains the same. We have the equation \( y = \sqrt[5]{x} \). Replacing x with -x, we get \( y = \sqrt[5]{-x} \). Since \( \sqrt[5]{-x} eq \sqrt[5]{x} \), the function is not symmetric about the y-axis.

- Test for symmetry about the x-axis

To test for symmetry about the x-axis, we replace y with -y in the equation and see if the equation remains the same. We have the equation \( y = \sqrt[5]{x} \). Replacing y with -y, we get \( -y = \sqrt[5]{x} \). Solving for y, we get \( y = -\sqrt[5]{x} \). Since \( -\sqrt[5]{x} eq \sqrt[5]{x} \), the function is not symmetric about the x-axis.

- Test for symmetry about the origin

To test for symmetry about the origin, we replace x with -x and y with -y in the equation and see if the equation remains the same. We have the equation \( y = \sqrt[5]{x} \). Replacing x with -x and y with -y, we get \( -y = \sqrt[5]{-x} \). Since \( \sqrt[5]{-x} = -\sqrt[5]{x} \), this simplifies to \( -y = -\sqrt[5]{x} \), which is equivalent to \( y = \sqrt[5]{x} \). Therefore, the function is symmetric about the origin.

Key concepts

x-intercept

The x-intercept is where the graph of a function crosses the x-axis. To find this point, we need to set the y-value to 0 and solve for x. For the given equation, \( y = \sqrt[5]{x} \), setting \( y \) to 0 gives us \( 0 = \sqrt[5]{x} \). This simplifies to \( x = 0 \). Thus, the x-intercept for this function is at (0,0).

y-intercept

The y-intercept is the point where a graph crosses the y-axis. To determine this point, we set x to 0 and solve for y. Given the equation, \( y = \sqrt[5]{x} \), by setting \( x \) to 0, we get \( y = \sqrt[5]{0} = 0 \). Therefore, the y-intercept is (0,0).

symmetry about the y-axis

A graph has symmetry about the y-axis if replacing \( x \) with \( -x \) in the equation results in the same equation. For \( y = \sqrt[5]{x} \), substituting \( -x \) gives us \( y = \sqrt[5]{-x} \). Since \( \sqrt[5]{-x} \) is not equal to \( \sqrt[5]{x} \), the graph is not symmetric about the y-axis.

symmetry about the x-axis

For symmetry about the x-axis, replacing \( y \) with \( -y \) should yield the original equation. Given \( y = \sqrt[5]{x} \), substituting \( -y \) gives us \( -y = \sqrt[5]{x} \). Solving for \( y \), we get \( y = -\sqrt[5]{x} \), which is not equivalent to \( \sqrt[5]{x} \). Hence, the graph is not symmetric about the x-axis.

symmetry about the origin

A function has symmetry about the origin if replacing both \( x \) with \( -x \) and \( y \) with \( -y \) results in the original equation. For \( y = \sqrt[5]{x} \), substituting \( -x \) for \( x \) and \( -y \) for \( y \) gives us \( -y = \sqrt[5]{-x} \). Since \( \sqrt[5]{-x} = -\sqrt[5]{x} \), we have \( -y = -\sqrt[5]{x} \), which simplifies to \( y = \sqrt[5]{x} \). Therefore, the graph is symmetric about the origin.