Problem 61 - College Algebra

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$ for the given function. $$f(x)=x^{2}-4 x+3$$

Step-by-Step Solution

Answer

The simplified difference quotient for the provided function is \(2x-4+h\).

Step 1: Substitute and Expand

First, substitute \(x+h\) into \(f(x)\) and expand that out to get \(f(x+h)\). Which leads to the function: \(f(x+h) = (x+h)^{2}-4(x+h)+3\). After simplifying, \(f(x+h)\) becomes: \(x^{2}+2xh+h^{2}-4x-4h+3\).

Step 2: Apply Difference Quotient Formula

Next, substitute \(f(x)\) and \(f(x+h)\) into the difference quotient formula: \(\frac{f(x+h)-f(x)}{h}\) Which after substituting gives us: \(\frac{x^{2} + 2xh + h^{2} - 4x - 4h + 3 - (x^{2} - 4x + 3)}{h}\)

Step 3: Simplify the Numerator

Then, simplify the numerator by cancelling the terms that appear on both the right and left sides of the - sign. This leaves us with: \(\frac{2xh + h^{2} - 4h}{h}\)

Step 4: Simplify the Fraction

Lastly, cancel out \(h\) in the numerator and denominator to simplify the fraction, which simplifies to: \(2x-4+h\)

Key Concepts

Function Simplification

Function simplification is an essential mathematical technique that helps in reducing complex expressions into simpler forms. It's crucial in solving problems more efficiently and clearly seeing the relationship between variables. In the context of difference quotients, simplifying the function can make it easier to calculate the slope of the secant line, or the average rate of change.
To simplify a function like the given polynomial, you need to start by expanding any expressions and then combining like terms. For instance, in the given problem, after substituting \(x+h\) into the function \(f(x)\), the expanded form was \(x^2 + 2xh + h^2 - 4x - 4h + 3\). Every time you expand or combine terms, ensure accuracy, as this is the foundation for further calculations.

To further simplify, identify and remove terms that cancel each other out. This way, you get rid of unnecessary complexity, making it easier to apply formulas and carry out computations efficiently.

Polynomial Function

Polynomial functions are a type of mathematical expression that involve sums of powers of a variable, most commonly noted as \(x\). Each term in the polynomial consists of a coefficient and a variable raised to a non-negative integer power. For example, the polynomial \(f(x) = x^2 - 4x + 3\) consists of three terms: \(x^2\), \(-4x\), and the constant term \(+3\).
Polynomials are simplistic yet powerful tools in algebra that form the building blocks for more complex expressions. They are characterized by their degree, which is the highest power of the variable in the polynomial – in this case, the degree is 2 because of \(x^2\).

Working with polynomials involves operations like addition, subtraction, multiplication, and division. Simplifying polynomial expressions often requires combining like terms and using basic arithmetic operations, providing a manageable representation of the function for solving equations or differentiating.

Difference Quotient Formula

The difference quotient formula, \(\frac{f(x+h)-f(x)}{h}\), is a crucial concept in calculus. It's used to find the average rate of change of a function over an interval. Essentially, it provides a way to approximate the derivative of a function, which represents the slope of the tangent to the curve at a particular point.
In the given exercise, the primary aim is to apply this formula to evaluate the change between \(f(x+h)\) and \(f(x)\). The function \(f(x)\) is a polynomial, and finding the difference \(f(x+h) - f(x)\) involves substituting \(x+h\) into the polynomial, expanding it, and then subtracting \(f(x)\) from it. As seen in the step-by-step solution, the difference quotient simplifies to \(2x - 4 + h\) after removing common terms.

• First, substitute \(x + h\) into the function, calculate \(f(x+h)\).
• Subtract \(f(x)\) from \(f(x+h)\), then divide the whole expression by \(h\).
• Simplify the resulting expression by canceling \(h\) when possible. This simplification shows the average rate of change and prepares for calculus methods such as differentiation.

Understanding this formula is foundational for grasping calculus concepts, especially when learning about derivatives.