Problem 55 - Elementary and Intermediate Algebra

Question

Find \(g(2)\) and \(g(3) .\) See Example 4. $$ g(x)=(x+1)^{2} $$

Step-by-Step Solution

Answer

\(g(2) = 9\) and \(g(3) = 16\).

Step 1: Understanding the Function

The function given is \(g(x) = (x+1)^2\), which means for any input \(x\), you need to plug it into the formula \((x+1)^2\).

Step 2: Calculate \(g(2)\)

To find \(g(2)\), substitute \(x = 2\) into the function. Thus, \(g(2) = (2+1)^2\). This simplifies to \(3^2\), which is 9.

Step 3: Calculate \(g(3)\)

To find \(g(3)\), substitute \(x = 3\) into the function. Thus, \(g(3) = (3+1)^2\). This simplifies to \(4^2\), which is 16.

Key Concepts

Function Evaluation

Function evaluation is an essential concept in mathematics. It involves determining the output of a function for a specific input. A function is a mathematical relation where each input is related to exactly one output. In the context of our exercise, we have the function \(g(x) = (x+1)^2\). When we evaluate a function, we replace the variable, usually \(x\), with a number. This allows us to see what the output will be for that particular input. For example:

• To find \(g(2)\), we replace \(x\) with 2.
• To find \(g(3)\), we replace \(x\) with 3.

By replacing the variable with specific numbers, we can calculate and analyze the results to understand the behavior of the function. This step is crucial in applying functions to real-world problems and in understanding mathematical operations.

Input Substitution

Input substitution is a straightforward yet powerful procedure in evaluating functions. It involves substituting a given number for the variable in a function. In our exercise with \(g(x) = (x+1)^2\), this substitution is a key step to finding what the function produces at certain points.Here's how it works:

• Identify the value we want to substitute, such as 2 or 3.
• Replace \(x\) with this value in the expression \((x+1)^2\).
• Solve the resulting expression to find the output.

For example, substituting \(x = 2\) transforms the expression into \((2+1)^2 = 3^2\), which equals 9. Similarly, substituting \(x = 3\) converts it to \((3+1)^2 = 4^2\), which equals 16. This method showcases the versatility of functions in mapping inputs to their outputs and is fundamental in understanding their application.

Squaring a Binomial

Squaring a binomial is a specific operation that expands an expression of the form \((a+b)^2\). Understanding this operation is very useful and frequently encountered in algebra.The expression \((x+1)^2\) in our function is a classic example of a binomial. Squaring this involves a simple rule:\[(a+b)^2 = a^2 + 2ab + b^2\]Applying this rule to our binomial:

• \(a = x\)
• \(b = 1\)

The expansion becomes:

• \(x^2 + 2 \cdot x \cdot 1 + 1^2\)
• Which simplifies to \(x^2 + 2x + 1\)

This step is particularly useful for both simplifying polynomials and verifying calculations. Understanding how to square a binomial not only helps in solving algebraic expressions but also builds a foundation for more complex mathematical concepts.