Problem 12 - Calculus

Question

For \(g(x)=x^{2}+2 x+3,\) find and simplify: (a) \(g(2+h)\) (b) \(g(2)\) (c) \(g(2+h)-g(2)\)

Step-by-Step Solution

Answer

(a) \(h^2 + 6h + 11\), (b) \(11\), (c) \(h^2 + 6h\)

Step 1: Evaluate \(g(2+h)\)

Start by substituting \(x = 2+h\) into the function \(g(x) = x^2 + 2x + 3\). Calculate: \[ g(2+h) = (2+h)^2 + 2(2+h) + 3 \]Expand and simplify:\[ (2+h)^2 = 4 + 4h + h^2 \]\[ 2(2+h) = 4 + 2h \]Thus, \[ g(2+h) = 4 + 4h + h^2 + 4 + 2h + 3 \]Combine like terms:\[ g(2+h) = h^2 + 6h + 11 \]

Step 2: Evaluate \(g(2)\)

Substitute \(x = 2\) in the function \(g(x) = x^2 + 2x + 3\):\[ g(2) = 2^2 + 2(2) + 3 \]Calculate each term:\[ 2^2 = 4 \]\[ 2(2) = 4 \]\[ g(2) = 4 + 4 + 3 = 11 \]

Step 3: Calculate \(g(2+h)-g(2)\)

From Step 1 and Step 2, use the simplified forms of \(g(2+h) = h^2 + 6h + 11\) and \(g(2) = 11\). Subtract:\[ g(2+h) - g(2) = (h^2 + 6h + 11) - 11 \]Simplify by eliminating \(+11 - 11 \):\[ g(2+h) - g(2) = h^2 + 6h \]

Key Concepts

Understanding Polynomial Functions

Polynomial functions are expressions composed of variables and coefficients, structured in a way that involves only non-negative integer exponents of the variable(s). These functions can have one or more terms like constant, linear, quadratic, cubic, and so on.
When you encounter a function like \[ g(x) = x^2 + 2x + 3 \]it is a quadratic polynomial function because the highest degree of variable \(x\) is 2. Quadratic functions often produce graphs that are parabolas.
A key aspect of polynomial functions is their degree, which is determined by the highest exponent of the variable present. For our function, the degree is 2, meaning it is a second-degree polynomial.

Exploring Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. They are the building blocks of polynomial functions.
A great example is the function given in the problem: \[ g(x) = x^2 + 2x + 3 \]In this expression:

• \(x^2\) is a term with a variable and an exponent.
• \(2x\) includes a coefficient multiplied by a variable.
• 3 is a constant term (no variable attached).

Algebraic expressions often need simplification to evaluate properly or to compare different forms, as seen in finding \(g(2+h)\) and simplifying the terms.

Mastering Function Simplification

Function simplification is the process of systematically reducing an expression to its simplest, most compact form. This process usually involves combining like terms and performing basic arithmetic operations.
In the exercise, after substituting \(2+h\) into the polynomial \(g(x)\), the expression \((2+h)^2 + 2(2+h) + 3\) was expanded and combined to yield \(h^2 + 6h + 11\).
Simplification helps in revealing the underlying structure of expressions and in eliminating redundancy, as demonstrated when \(g(2+h) - g(2)\) resulted in \(h^2 + 6h\) by canceling out the common terms. This can be crucial for understanding the behavior of functions, especially when calculating differences or changes.