If \(f(x)=x^{2}-2,\) then \(f(x+h)=?\) F. \(x^{2}+h^{2}\) G. \(x^{2}-2+h\) H. \(x^{2}+h^{2}-2\) J. \(x^{2}+2 x h+h^{2}\) K. \(x^{2}+2 x h+h^{2}-2\)

When working with algebraic expressions, understanding how to expand binomials is crucial. A binomial is a polynomial with two terms, typically written in the form \(a + b\).

The binomial expansion refers to expressing the power of a binomial, \( (a+b)^n \), as a sum of terms involving coefficients, the variable \( a \) raised to different powers, and the variable \( b \) raised to other powers. These coefficients can be determined by Pascal's Triangle or the binomial theorem, which states that:\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k \]
For instance, when we wanted to find \( (x+h)^2 \), the binomial expansion formula gave us \( x^2 + 2xh + h^2 \), a neat and systematic way to expand and simplify the expression.

Algebraic functions are functions that can be constructed using algebraic operations: addition, subtraction, multiplication, division, and taking roots. These functions are defined by polynomials or ratios of polynomials. In our example, \( f(x) = x^2 - 2 \) is an algebraic function, specifically a polynomial function, which is a single-variable function that involves only powers of that variable.

It's important to notice that algebraic functions like our \( f(x) \) take an input \( x \) and produce an output based on the operation defined by the function's rule. When we substitute \( x+h \) into \( f(x) \) to find \( f(x+h) \) as in our original problem, we are essentially manipulating the input of the function, which in turn alters the output according to the rule set by the function's algebraic form. Understanding function notation and how to work with inputs is fundamental to mastering algebra and calculus.

Function transformation involves changing the original function's rule to produce a shifted, stretched, or reflected graph without altering its fundamental shape. In algebra, we often achieve this by applying arithmetic operations directly to the function's formula, which is precisely what we are doing by finding \( f(x+h) \).

By replacing \( x \) in the function with \( x+h \) we are applying a horizontal shift to the function \( f(x) \). If \( h \) is positive, the function's graph shifts \( h \) units to the left; if negative, the graph moves \( h \) units to the right. While our main exercise did not concentrate on graphing these transformations, it provides the foundation for understanding how functions are altered and how these changes are reflected algebraically.

Mastering function transformation is a powerful tool, enabling the prediction of changes to the graph of a function without having to plot multiple points. It's a step toward visualizing mathematical concepts and enhances the understanding of how mathematics can model real-world phenomena.