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Function substitution is a foundational concept in calculus and algebra that helps you evaluate a function at different points. In this process, you replace the current input value of a function with another expression to compute a new output.
For example, consider the function given in the exercise: \( f(x) = \frac{1}{x^2} \). To find the function at a new point \( x+h \), substitute \( x \) with \( x+h \).

• Substitution: \( f(x+h) = \frac{1}{(x+h)^2} \)
• Initial Function: \( f(x) = \frac{1}{x^2} \)

This substitution is crucial when dealing with difference quotients, as it allows us to calculate the change in function values as their inputs change. Practicing function substitution helps you become comfortable with manipulating functions to find derivatives and understand function behavior.

The difference quotient, \( \frac{f(x+h) - f(x)}{h} \), is an essential concept in introductory calculus. It represents the average rate of change of the function \( f(x) \) over the interval \( x \) to \( x+h \). Simplifying the difference quotient gives insight into the derivative, which defines the instantaneous rate of change.To simplify, begin by substituting the computed function values, \( f(x) \) and \( f(x+h) \), into the quotient:

• \( \frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h} \)

Next, find a common denominator for \( \frac{1}{(x+h)^2} \) and \( \frac{1}{x^2} \). Multiply to form:

• Numerator: \( \frac{x^2 - (x+h)^2}{x^2(x+h)^2} \)

Finally, simplify by expanding and combining like terms to neatly and accurately reduce to the simplest form. Simplification not only improves clarity but also prepares equations for further calculus processes like differentiation.

Rational expressions involve fractions with polynomials in the numerator and the denominator. Simplifying these expressions can be broken into manageable steps that often involve finding a common denominator and canceling out terms when possible.To simplify a rational expression such as \( \frac{1}{(x+h)^2} - \frac{1}{x^2} \), start by identifying a common denominator. In this example, it is \( x^2(x+h)^2 \).

• Rewriting: \( \frac{x^2}{x^2(x+h)^2} - \frac{(x+h)^2}{x^2(x+h)^2} = \frac{x^2 - (x+h)^2}{x^2(x+h)^2} \)
• Expand \( (x+h)^2 \): \( x^2 + 2xh + h^2 \)
• Simplify the numerator: \( x^2 - (x^2 + 2xh + h^2) = -2xh - h^2 \)

After expanding and simplifying, factor out the common terms. In this case, factor \( h \) from the numerator. Then cancel \( h \) from both the numerator and the denominator when possible, resulting in:

• Final Simplification: \( \frac{-2x - h}{x^2(x+h)^2} \)

These steps make rational expressions manageable and ready for subsequent calculations, such as derivative analysis in calculus.