Problem 78 - A Graphical Approach to College Algebra

Question

For each function find \(f(x+h)\) and \(f(x)+f(h)\). $$f(x)=x^{3}$$

Step-by-Step Solution

Answer

\(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\); \(f(x) + f(h) = x^3 + h^3\).

Step 1: Identify the Function

The given function in the problem is \(f(x) = x^3\). This is a cubic function.

Step 2: Calculate \(f(x+h)\)

Substitute \((x+h)\) into the function in place of \(x\). Thus, \(f(x+h) = (x+h)^3\).

Step 3: Expand \((x+h)^3\)

Use the binomial expansion \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\). This results from expanding the cubic expression using the binomial theorem.

Step 4: Calculate \(f(x) + f(h)\)

Substitute \(x\) and \(h\) into the function separately. Therefore, \(f(x) = x^3\) and \(f(h) = h^3\). Now, compute \(f(x) + f(h) = x^3 + h^3\).

Step 5: Compare \(f(x+h)\) and \(f(x) + f(h)\)

You have \(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\) and \(f(x) + f(h) = x^3 + h^3\). Notice how extra terms \(3x^2h + 3xh^2\) are present in \(f(x+h)\) but not in \(f(x) + f(h)\).

Key Concepts

Binomial Expansion

Binomial expansion is a powerful tool in algebra to simplify expressions of the form y \− x\((a + b)^n\). In the case of our problem, the expression \((x + h)^3\) is expanded using the binomial theorem. The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Applying this to \((x + h)^3\), we get:

• \(x^3\) - from the term where \(k = 0\)
• \(3x^2h\) - from the term with \(k = 1\)
• \(3xh^2\) - where \(k = 2\)
• \(h^3\) - from when \(k = 3\)

This expansion includes calculating the required coefficients using combinations, and it allows you to write the polynomial in a simpler extended form. Thus, binomial expansion is essential for deriving expressions from powers, simplifying polynomials, and solving equations easily. In this exercise, it helps illustrate how \(f(x+h)\) contains more terms than the sum \(f(x) + f(h)\).

Function Notation

Function notation is a mathematical symbol used to define a function. It describes situations where one quantity depends on another.In this exercise, \(f(x)\)represents a function of \(x\). This means that the outcome of the function depends on the value of \(x\).When the exercise asks for \(f(x+h)\), it asks you to evaluate the function by replacing every instance of \(x\) with \((x + h)\). Similarly, \(f(x) + f(h)\) means you need to evaluate the function for each of \(x\) and \(h\) separately, and then add the results. Function notation is crucial because it provides a clear way to express the relationship between variables in algebra. It helps you track changes in inputs and their corresponding outcomes with precision, which is especially useful when comparing different expressions of the same function.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators such as +, −, ·, and ÷. These expressions form the basis for operations in algebra and provide a means to communicate mathematical ideas and solve equations. In our specific exercise, \(x^3\), \(h^3\), and the expanded expression \((x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\) are algebraic expressions. Understanding how to manipulate these expressions through processes such as expansion or simplification is core to solving algebraic problems. The ability to recognize equivalent expressions and compare their forms helps us find solutions and understand relationships between variables. For example, seeing the difference between \(f(x+h)\) and \(f(x) + f(h)\) illustrates how functions behave differently when combined directly versus when evaluated separately.