Problem 71 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

(a) \(f(x+h) = x^2 + 2xh + h^2 - 4\); (b) \(f(x) + f(h) = x^2 + h^2 - 8\).

Step 1: Find f(x+h)

To find \(f(x+h)\), we substitute \(x+h\) into the original function \(f(x) = x^2 - 4\). This gives us: \[ f(x+h) = (x+h)^2 - 4 \]. Next, expand \((x+h)^2\): \[ (x+h)^2 = x^2 + 2xh + h^2 \]. Substitute back into the function: \[ f(x+h) = x^2 + 2xh + h^2 - 4 \]. Therefore, \(f(x+h) = x^2 + 2xh + h^2 - 4\).

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

We start with the expressions for \(f(x)\) and \(f(h)\). Given \(f(x) = x^2 - 4\), we need \(f(h)\) which is found by replacing \(x\) with \(h\): \[ f(h) = h^2 - 4 \]. Now sum \(f(x)\) and \(f(h)\): \[ f(x) + f(h) = (x^2 - 4) + (h^2 - 4) \]. Simplify the expression: \[ f(x) + f(h) = x^2 + h^2 - 8 \].

Key Concepts

Function Operations

Understanding function operations is essential in precalculus. These operations involve combining and manipulating functions through various means such as addition, subtraction, multiplication, division, and composition.
For example, when a problem asks for \(f(x+h)\) or \(f(x) + f(h)\), you are dealing with the function operation of substitution and addition.

• Substitution involves replacing the variable in the function with a different expression or number, like substituting \(x+h\) into \(f(x) = x^2 - 4\), resulting in \((x+h)^2 - 4\).
• Addition of functions is another operation, seen when you add two separate functions together, such as \(f(x) + f(h)\).

This allows us to analyze how functions behave under different circumstances, helping to strengthen our understanding of the function’s overall behavior. Function operations are the building blocks towards more complex mathematical concepts.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations, such as addition and multiplication. They are a fundamental part of algebra and precalculus. In solving problems, understanding how to manipulate these expressions is crucial.
To solve for \(f(x+h)\), we expanded the algebraic expression \((x+h)^2\):

• This involves applying the distributive property, resulting in \(x^2 + 2xh + h^2\).

Managing these expressions involves:

• Combining like terms, as done when simplifying \(f(x) + f(h)\), resulting in \(x^2 + h^2 - 8\).
• Using foundational skills like expansion (expanding \((x+h)^2\) to its full form) and distribution.

Mastering these skills is vital as they are commonly used in simplifying expressions and solving equations throughout mathematics.

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers. In our example, \(f(x) = x^2 - 4\) is a polynomial function of degree 2, often called a quadratic function.
These functions have specific characteristics:

• The degree of the function determines its shape and the number of its roots. Quadratic functions, for example, generally create a parabolic shape.
• Understanding how changes in the function, like \(f(x+h)\), affect its form is critical in advanced mathematics.

Operations on polynomials, such as addition seen in \(f(x) + f(h)\), help deepen our understanding of how polynomials are structured and their behavior through transformations.
These insights are foundational for analyzing more complex algebraic structures and systems and used frequently across higher mathematics disciplines.