91Ó°ÊÓ

Function substitution can seem tricky, but it's quite straightforward once you understand the basic idea. Let's break it down:

In this exercise, we first need to substitute a new expression into our given function. The original function here is \(f(x) = x^2\). To find \(f(x+h)\), we replace every occurrence of \(x\) in \(f(x)\) with \(x+h\).

So we substitute \(x+h\) into \(f(x)\):

\[ f(x + h) = (x + h)^2 \]

This substitution step is essential because we are re-evaluating the function based on a shifted input (from \(x\) to \(x + h\)). It's a key move in many calculus problems, especially those involving limits and derivatives.

Next, we need to expand the expression \( (x + h)^2 \) following the binomial expansion formula. The binomial expansion formula for \( (a + b)^2 \) is: \[ (a + b)^2 = a^2 + 2ab + b^2 \]

Applying this formula to our case where \( a = x \) and \( b = h \), we get:

\[ (x + h)^2 = x^2 + 2xh + h^2 \]

Binomial expansion takes a product of sums and expands it into a sum of products. It's executed by distributing the terms, ensuring all possible combinations are covered. This step lays the foundation for further simplification.

Finally, we need to simplify the expression by combining like terms. After substituting and expanding, we have:

\[ f(x+h) - f(x) = (x^2 + 2xh + h^2) - x^2 \]

Combining like terms involves canceling out terms that are identical except for their coefficients. Here, the \( x^2 \) terms cancel each other out:

\[ x^2 + 2xh + h^2 - x^2 = 2xh + h^2 \]

By removing these terms, we obtain the simplified difference:

\[ f(x+h) - f(x) = 2xh + h^2 \]

Breaking down expressions into simpler parts not only makes them easier to understand but also prepares them for further mathematical operations, like taking limits or finding derivatives. Combining like terms is a crucial skill in algebra that helps in simplifying and solving equations efficiently.