(a) A function f is said to be odd if the equation...

(a) A function $f$ is said to be odd if the equation $f(-x)=-f(x)$ is satisfied by all values of $x$ in the domain of $f .$ Show that if $(x, y)$ is a point on the graph of an odd function $f,$ then the point $(-x,-y)$ is also on the graph. (This implies that the graph of an odd function must be symmetric about the origin.) (b) Show that each function is odd by computing $f(-x)$ as well as $-f(x)$ and then noting that the two expressions obtained are equal. (i) $f(x)=x^{3}$ (iii) $f(x)=|x| /\left(x+x^{7}\right)$ (ii) $f(x)=-2 x^{5}+4 x^{3}-x$

(a) A function $f$ is said to be odd if the equation $f(-x)=-f(x)$ is satisfied by all values of $x$ in the domain of $f .$ Show that if $(x, y)$ is a point on the graph of an odd function $f,$ then the point $(-x,-y)$ is also on the graph. (This implies that the graph of an odd function must be symmetric about the origin.)
(b) Show that each function is odd by computing $f(-x)$ as well as $-f(x)$ and then noting that the two expressions obtained are equal.
(i) $f(x)=x^{3}$
(iii) $f(x)=|x| /\left(x+x^{7}\right)$
(ii) $f(x)=-2 x^{5}+4 x^{3}-x$

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Key Concepts

Odd Function

An odd function is defined by the property f(-x) = -f(x) for every x in its domain. This definition emphasizes that the output for the input -x is the negative of the output for x, which is a foundational concept in analyzing function behavior.

Origin Symmetry

One of the key features of odd functions is that their graphs are symmetric about the origin. This implies that for every point (x, y) on the graph of an odd function, the corresponding point (-x, -y) will also lie on the graph, revealing a specific type of reflective symmetry inherent in the function’s structure.

Substitution Method for Verification

To verify whether a given function is odd, the standard approach is to substitute -x into the function to obtain f(-x) and then compute -f(x). Comparing these two expressions algebraically confirms if the function meets the odd function criterion. This method is central to proving oddness in functions.

Handling Piecewise Functions and Absolute Values

When dealing with functions that involve absolute values or piecewise definitions, particularly in verifying oddness, careful attention must be given to how the properties of absolute values affect the function. Evaluating f(-x) in these contexts ensures that the odd function condition holds true across all segments of the function's domain.

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