Question

11
select the correct answer.
which statement is true about function f, which is shown in the graph?
f(x) = -8x⁵ + 4x³ + 5x
graph of a function on a coordinate plane with x-axis from -5 to 5 and y-axis from -5 to 5
a. function f is even.
b. function f is neither even nor odd.
c. function f is both even and odd.
d. function f is odd.

Explanation:

Step1: Recall odd/even function definitions

A function \( f(x) \) is odd if \( f(-x) = -f(x) \), even if \( f(-x) = f(x) \).

Step2: Compute \( f(-x) \) for \( f(x) = -8x^5 + 4x^3 + 5x \)

Substitute \( -x \) into \( f(x) \):
\( f(-x) = -8(-x)^5 + 4(-x)^3 + 5(-x) \)
Simplify each term:
\( (-x)^5 = -x^5 \), so \( -8(-x)^5 = 8x^5 \);
\( (-x)^3 = -x^3 \), so \( 4(-x)^3 = -4x^3 \);
\( 5(-x) = -5x \).

Thus, \( f(-x) = 8x^5 - 4x^3 - 5x \).

Step3: Compare \( f(-x) \) with \( -f(x) \)

Compute \( -f(x) \):
\( -f(x) = -(-8x^5 + 4x^3 + 5x) = 8x^5 - 4x^3 - 5x \).

Since \( f(-x) = -f(x) \), \( f(x) \) is odd. Also, the graph is symmetric about the origin (a property of odd functions), confirming this.

Answer:

D. Function \( f \) is odd.