Question

given \\(f(x) = -4x^6 + 3|x|\\), answer the following:

part: 0 / 3

part 1 of 3

(a) find \\(f(-x)\\).

\\(f(-x) = \\)

Explanation:

Identify the given function

We are given the function:

$$f(x) = -4x^6 + 3|x|$$

Substitute \(-x\) for \(x\)

To find \(f(-x)\), we substitute \(-x\) into every occurrence of \(x\) in the function definition:

$$f(-x) = -4(-x)^6 + 3|-x|$$

Simplify the even power term

Since any negative number raised to an even power becomes positive:

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

Therefore:

$$-4(-x)^6 = -4x^6$$

Simplify the absolute value term

By definition, the absolute value of a number and its negative are equal:

$$|-x| = |x|$$

Therefore:

$$3|-x| = 3|x|$$

Combine the simplified terms

Combining the simplified terms gives:

$$f(-x) = -4x^6 + 3|x|$$

Answer:

Given \(f(x) = -4x^6 + 3|x|\), answer the following:

(a) Find \(f(-x)\).

\(f(-x) =\) <blank>\(-4x^6 + 3|x|\)</blank>