Understanding f(x² – 2) = (x² – 2)² – 1: A Complete Breakdown

Have you ever encountered a function defined in a surprising but elegant algebraic form like f(x² – 2) = (x² – 2)² – 1? If so, you’re not alone—this function offers a clever way to simplify complex expressions while revealing deeper insights into polynomial relationships. In this SEO-optimized article, we’ll explore the transformation, simplify the expression, and uncover the elegance behind f(x² – 2) and its expanded form f(x² – 2) = x⁴ – 4x² + 3.

Understanding the Context

What Does f(x² – 2) = (x² – 2)² – 1 Mean?

At first glance, f(x² – 2) appears cryptic, but breaking it down reveals a piecewise function logic based on substitution. When we see f(u) where u = x² – 2, replacing u in the functional form:

> f(u) = u² – 1

Substituting back:

Solved 1. f(x)=x4−4x2+5x−4 f(x)=x−2cosx+2 f(x)=−2x3+4x2+3x−4 | Chegg.com

Image Gallery

Solved 1. f(x)=x4−4x2+5x−4 f(x)=x−2cosx+2 f(x)=−2x3+4x2+3x−4 | Chegg.com
Solved Question | Chegg.com
Solved f(x) = -2x + 1 f(x) = -2x^2 - 3x f(x) = -1/4 x^4 | Chegg.com
Solved Find f-1(x).f(x)=2xx-4,x≠4f-1(x)=State any | Chegg.com
Solved F(x) = x - 2/x^2 - 4 f(x) = x - 3/x^2 - 9 f(x) = | Chegg.com

Key Insights

> f(x² – 2) = (x² – 2)² – 1

This reveals that f transforms its input by squaring it and subtracting 1. But what’s the functional shape? Let’s expand and simplify.

Step-by-Step Simplification: From (x² – 2)² – 1 to x⁴ – 4x² + 3

We begin with:

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Final Thoughts

> (x² – 2)² – 1

Using the algebraic identity (a – b)² = a² – 2ab + b²:

> (x² – 2)² = (x²)² – 2·x²·2 + 2² = x⁴ – 4x² + 4

Now subtract 1:

> x⁴ – 4x² + 4 – 1 = x⁴ – 4x² + 3

So finally:

> f(x² – 2) = x⁴ – 4x² + 3

Why This Matters: Simplifying Functional Expressions

Expressions like f(x² – 2) often appear in algebra, calculus, and even physics when modeling transformations. By simplifying f(x² – 2), we uncover its true degree and coefficients — in this case, a quartic function in x.