Analyze transformations of quadratic and absolute-value functions and identify even/odd functions.
Determine whether a transformed quadratic or absolute-value function is even
Problem
Determine whether \(f(x)=x^{2}+3\) is even. Compute and simplify \(f(-x)\), compare it explicitly with \(f(x)\), state the conclusion, and connect the algebraic test to symmetry about the \(y\text{-}\text{axis}\).
Big Picture
What this problem is really about
Evenness is an identity that must hold for every input, not a check at one point. We’ll substitute negative x into the complete rule, simplify the squared term carefully, compare the result with f of x, and translate the equality into y-axis symmetry.
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