AI Tutor
If f(x)=x2βˆ’3xf(x) = x^2 - 3x, what is f(x+h)f(x + h)? A common trap is writing x2βˆ’3x+hx^2 - 3x + h, treating function notation like a label rather than an active recipe.
The notation f(input)f(\text{input}) means that every single instance of the independent variable xx in the original definition must be completely replaced by whatever expression sits inside the parentheses.
How do we handle more complicated polynomial inputs without making sign or expansion errors?

Evaluating Polynomial Expressions

Let us evaluate f(2aβˆ’1)f(2a - 1) for the quadratic function f(x)=3x2βˆ’4x+5f(x) = 3x^2 - 4x + 5.
First, create an open shell with parentheses for every xx, and substitute (2aβˆ’1)(2a - 1): f(2aβˆ’1)=3(2aβˆ’1)2βˆ’4(2aβˆ’1)+5f(2a-1) = 3(2a - 1)^2 - 4(2a - 1) + 5
Always expand binomial squares carefully: (2aβˆ’1)2=4a2βˆ’4a+1(2a - 1)^2 = 4a^2 - 4a + 1, never just 4a2+14a^2 + 1. Forgetting the middle cross-term βˆ’4a-4a is the most frequent algebraic mistake in polynomial evaluation.
What happens when our function is a fraction, and we need to evaluate the rate of change between two points?

The Difference Quotient and Rational Functions

In the 1630s, Pierre de Fermat laid the groundwork for calculus by comparing a curve's height at xx versus x+hx+h. This expression is the difference quotient: f(x+h)βˆ’f(x)h\frac{f(x+h) - f(x)}{h}
The difference quotient represents the slope of the secant line passing through (x,f(x))(x, f(x)) and (x+h,f(x+h))(x+h, f(x+h)), measuring the average rate of change over an interval of width hh.