Problem 74 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the given function

We have the function \(f(x) = x^3\) given in the exercise. This is a cubic function which is continuous and differentiable for all real numbers.

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

In order to find \(f(x+h)\), substitute \(x+h\) into the function \(f(x) = x^3\). \[f(x+h) = (x+h)^3\] Now, use the binomial expansion to expand this cube:\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\] Thus, \(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\).

Step 3: Find \(f(x) + f(h)\)

Next, calculate \(f(x) + f(h)\) by evaluating \(f(h)\) and adding it to \(f(x)\).Start by computing \(f(h) = h^3\) since we plug \(h\) into the function.Thus, \(f(x) + f(h) = x^3 + h^3\).

Key Concepts

Binomial Expansion

The binomial expansion is a critical concept in understanding expressions raised to a power, such as \((x + h)^3\) in our case. Here, we aim to expand a binomial expression like \((a + b)^n\) into a sum of terms. Each term is a product of binomial coefficients, powers of \(a\) and \(b\). The binomial theorem states:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

For our specific problem, where \(n = 3\), we expand \((x + h)^3\) using:

• \(\binom{3}{0} x^3 = x^3\)
• \(\binom{3}{1} x^2 h = 3x^2 h\)
• \(\binom{3}{2} x h^2 = 3x h^2\)
• \(\binom{3}{3} h^3 = h^3\)

This simplifies to \(x^3 + 3x^2h + 3xh^2 + h^3\). The concept helps break down complex expressions into manageable components, easing evaluation and analysis.

Function Evaluation

Evaluating functions is a vital math skill that allows us to determine the output for a given input. For a function \(f(x)\), this involves substituting a specific value or expression for \(x\) and then simplifying.
In the given exercise, we have the function \(f(x) = x^3\). To find \(f(x+h)\), we replace every occurrence of \(x\) with \(x+h\), yielding \((x+h)^3\). Using techniques such as binomial expansion, we simplify it to \(x^3 + 3x^2h + 3xh^2 + h^3\).
Similarly, to find \(f(h)\), substitute \(h\) into the function: \(f(h) = h^3\).
This process allows us to transform and evaluate expressions effectively with variables or numbers, providing insight into how functions behave under different conditions.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole-number powers and coefficients. They are classified based on their highest degree. In this exercise, \(f(x) = x^3\) is a cubic function—a type of polynomial function with a degree of three.
These functions are continuous and smooth, making them predictable and easy to work with. Polynomials can model many real-world situations because their properties are straightforward to analyze.

• A cubic function like \(x^3\) is symmetric about the origin, reflecting its odd-degree nature.
• Polynomial functions can be added, subtracted, multiplied, and divided (except by zero).
• They can be evaluated at any point, giving insights into their behavior and graph characteristics, like intercepts and turning points.

Understanding these types of functions allows students to solve complex problems and interpret various patterns in mathematics effectively.