Understanding Odd Functions: Proving the Oddness of a Function f(x) through Substitution and Simplification

Show f(x) is odd

To show that a function f(x) is odd, we need to prove that f(x) satisfies the definition of an odd function

To show that a function f(x) is odd, we need to prove that f(x) satisfies the definition of an odd function.

Definition of an Odd Function:
A function f(x) is said to be odd if for every value of x in its domain, f(-x) = -f(x).

To prove that f(x) is odd, we will substitute -x into the function and show that it equals the negative of f(x).

Let’s start by substituting -x into the function f(x):

f(-x) = -2(-x)^3 + 3(-x) + 4

Now, let’s simplify this expression.

Using the property (-a)^n = -a^n for odd values of n:

f(-x) = -2(-x)^3 + 3(-x) + 4
= -2(-(x)^3) + 3(-x) + 4
= -2(-x^3) + 3(-x) + 4

Expanding the exponent:

f(-x) = 2x^3 + 3(-x) + 4
= 2x^3 – 3x + 4

Now, let’s compare this result with -f(x):

-f(x) = -[-2x^3 + 3x + 4]
= 2x^3 – 3x – 4

Comparing the two expressions, we can see that f(-x) = -f(x), which satisfies the definition of an odd function.

Therefore, we have shown that f(x) is an odd function.

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