Question

determine whether the function is even, odd, or neither. then determine whether the functions graph is symmetric with respect to the y-axis, the origin, or neither.

(f(x) = x^6 + x^{10} + 2)

determine whether the function is even, odd, or neither. choose the correct answer below.

neither
odd
even

determine whether the graph of the function is symmetric with respect to the y-axis, the origin, or neither. select all that apply.

origin
neither
y-axis

Explanation:

Evaluate the function at negative x

To determine if the function \(f(x) = x^6 + x^{10} + 2\) is even, odd, or neither, we evaluate \(f(-x)\) by substituting \(-x\) for \(x\):

$$ f(-x) = (-x)^6 + (-x)^{10} + 2 $$

Simplify the expression

Since any negative number raised to an even power becomes positive, we simplify the terms:

$$ (-x)^6 = x^6 $$

$$ (-x)^{10} = x^{10} $$

Thus, the simplified function is:

$$ f(-x) = x^6 + x^{10} + 2 $$

Compare with the original function

We compare our result for \(f(-x)\) with the original function \(f(x)\):

$$ f(-x) = f(x) $$

Since \(f(-x) = f(x)\), the function is even.

Determine the graphical symmetry

An even function is symmetric with respect to the y-axis, whereas an odd function is symmetric with respect to the origin. Since \(f(x)\) is an even function, its graph is symmetric with respect to the y-axis.

Answer:

Question 1

• neither
• odd
• even (Correct answer)

Question 2

• origin
• neither
• y-axis (Correct answer)