Determine analytically if the functions are even odd or neit

Determine analytically if the functions are even, odd, or neither. f(x) = 4x^2 - x + 12 f (x) = x^3 + x -15

Solution

A function f is even if the graph of f is symmetric with respect to the y-axis. Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f.

A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if

f(-x) = -f(x) for all x in the domain of f.

a) f(x) = 4x^2 - x +12

f(-x) = 4(-x)^2 -(-x) +12 = 4x^2 +x +12

So, f(-x) is neither equal to f(x) nor equal to -f(x)

function is neither even nor odd

b) f(x) = x^3 + x -15

f(-x) = (-x)^3 + (-x) -15 = -x^3 -x -15

So, f(-x) is neither equal to f(x) nor equal to -f(x)

function is neither even nor odd