Problem 75 - A Graphical Approach to College Algebra

Question

For each function find \(f(x+h)\) and \(f(x)+f(h)\). $$f(x)=x^{2}-4$$

Step-by-Step Solution

Answer

\(f(x+h) = x^2 + 2xh + h^2 - 4\), and \(f(x)+f(h) = x^2 + h^2 - 8\).

Step 1: Find \(f(x+h)\)

To find \(f(x+h)\), we substitute \(x+h\) into the function \(f(x) = x^2 - 4\) wherever we see \(x\).Thus, \(f(x+h) = (x+h)^2 - 4\).

Step 2: Expand \((x+h)^2\)

Now, let's expand the term \((x+h)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\):\[(x+h)^2 = x^2 + 2xh + h^2\].

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

Substituting the expanded form into \(f(x+h)\), we get:\[f(x+h) = x^2 + 2xh + h^2 - 4\].

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

Compute \(f(x) + f(h)\) individually:\[f(x) = x^2 - 4\] and \[f(h) = h^2 - 4\].Therefore, \(f(x) + f(h) = (x^2 - 4) + (h^2 - 4)\).

Step 5: Simplify \(f(x) + f(h)\)

Combine the terms from the expression:\[f(x) + f(h) = x^2 + h^2 - 8\].

Key Concepts

Polynomial Functions

Polynomial functions are foundational in algebra and consist of variables raised to whole number powers. The form of these functions can vary, like linear or quadratic.
The function in our example, \(f(x) = x^2 - 4\), is quadratic. Quadratic means the highest power of \(x\) is squared.
Such functions may be simple to evaluate, yet they require careful attention regarding how each term behaves as a whole.

• Terms: Expressions like \(x^2\) or constant numbers such as 4.
• Quadratic Function: Takes the format \(ax^2 + bx + c\), with \(a\) not equal to 0. Here, \(a=1\), \(b=0\), and \(c=-4\).
• Purpose: Polynomial functions can model various real-world phenomena and allow complex calculations like differentiating motion or optimization problems.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them easier to understand and solve.
It includes expanding, factoring, simplifying terms, and combining like terms. Let's delve into how this method works in our previous solution steps.
For instance, when evaluating \(f(x+h)\), we need to expand \((x+h)^2\). This step requires the use of the binomial expansion formula:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• Therefore, \((x+h)^2 = x^2 + 2xh + h^2\)

Here, algebraic manipulation simplifies analysis and helps keep track of all parts of the polynomial as they interact. By distributing and combining terms, one can manage complex polynomial expressions effectively. Emphasizing clarity helps ensure that no detail is lost in arithmetic complexity.

Substitution Method

The substitution method is a technique where you replace variables in an equation or function with specific values or expressions. It is invaluable when solving problems involving functions like polynomial evaluations.
In our exercise, consider how we used substitution:

• To find \(f(x+h)\), substitute \(x+h\) for \(x\) in the function \(f(x) = x^2 - 4\).
• Determine \(f(h)\) by replacing \(x\) with \(h\), giving \(h^2 - 4\).

Substitution simplifies complex polynomial expressions by breaking them into smaller, more manageable parts. This method is handy for function evaluation, allowing us to observe how changes in input affect output, crucial for mathematics and applied sciences.