Simplify the difference quotient ƒ(x+h)-ƒ(x)/h
ƒ(x) = 4x-3

Verified step by step guidance

1. Step 1: Start by substituting the function \( f(x) = 4x - 3 \) into the difference quotient formula \( \frac{f(x+h) - f(x)}{h} \).
2. Step 2: Calculate \( f(x+h) \) by replacing \( x \) with \( x+h \) in the function \( f(x) = 4x - 3 \). This gives \( f(x+h) = 4(x+h) - 3 \).
3. Step 3: Expand \( f(x+h) = 4(x+h) - 3 \) to get \( 4x + 4h - 3 \).
4. Step 4: Substitute \( f(x+h) = 4x + 4h - 3 \) and \( f(x) = 4x - 3 \) into the difference quotient: \( \frac{(4x + 4h - 3) - (4x - 3)}{h} \).
5. Step 5: Simplify the expression in the numerator: \( (4x + 4h - 3) - (4x - 3) = 4h \). The difference quotient becomes \( \frac{4h}{h} \).

Key Concepts

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as (ƒ(x+h) - ƒ(x)) / h, where h represents a small change in x. This concept is fundamental in calculus as it leads to the definition of the derivative, which measures the instantaneous rate of change.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, we evaluate the function ƒ(x) = 4x - 3 at two points: x and x + h. Understanding how to correctly substitute values into a function is crucial for simplifying expressions like the difference quotient.

Algebraic Simplification

Algebraic simplification is the process of manipulating an expression to make it easier to work with or to reveal its underlying structure. This includes combining like terms, factoring, and reducing fractions. In the context of the difference quotient, simplifying the expression after substituting the function values is essential to arrive at a clearer form, which can then be analyzed further.