Retaining the Concepts. If f(x) = 4x² - 5x - 2, find [f(x + h) - f(x)]/h, h ≠ 0

Verified step by step guidance

1. Start by writing the given function: $f\left(x\right)=\; 4x{x}^2-\; 5x\; -\; 2$.
2. Find the expression for $f(x\; +\; h)$ by substituting $x\; +\; h$ into the function: $f(x\; +\; h)\; =\; 4(x\; +\; h)^2\; -\; 5(x\; +\; h)\; -\; 2$.
3. Expand the squared term and distribute the constants: expand $(x\; +\; h)^2$ to $x^2\; +\; 2xh\; +\; h^2$, then multiply by 4, and distribute -5 over $(x\; +\; h)$.
4. Write the difference quotient expression: $\backslash frac\{f(x\; +\; h)\; -\; f(x)\}\{h\}$, substituting the expanded form of $f(x\; +\; h)$ and the original $f(x)$.
5. Simplify the numerator by combining like terms and then factor out $h$ from the numerator to cancel with the denominator, remembering that $h\; \backslash neq\; 0$.

Key Concepts

Function Notation and Evaluation

Function notation, such as f(x), represents a rule that assigns each input x to an output. Evaluating f(x + h) means substituting x + h into the function in place of x, which is essential for understanding how the function changes with small increments.

Difference Quotient

The difference quotient [f(x + h) - f(x)]/h measures the average rate of change of the function over the interval from x to x + h. It is foundational in calculus as it approximates the derivative and helps analyze how functions behave locally.

Polynomial Simplification

Simplifying expressions involving polynomials requires expanding terms, combining like terms, and factoring when possible. Mastery of these algebraic manipulations is necessary to simplify the difference quotient and express it in a reduced form.