Problem 25 - Algebra and Trigonometry

Question

Use the function to evaluate the indicated expressions and simplify. $$ f(x)=x^{2}+1 ; \quad f(x+2), f(x)+f(2) $$

Step-by-Step Solution

Answer

\(f(x+2) = x^2 + 4x + 5\); \(f(x) + f(2) = x^2 + 6\).

Step 1: Evaluate f(x+2)

To find \(f(x+2)\), substitute \(x+2\) for \(x\) in the function \(f(x) = x^2 + 1\). \[f(x+2) = (x+2)^2 + 1\]Simplify the expression by expanding the square:\[(x+2)^2 = x^2 + 4x + 4\]Therefore,\[f(x+2) = x^2 + 4x + 4 + 1 = x^2 + 4x + 5\]

Step 2: Evaluate f(x) + f(2)

First, evaluate \(f(x)\) and \(f(2)\) separately. Using the function \(f(x) = x^2 + 1\), we have:- \(f(x) = x^2 + 1\)- For \(f(2)\):\[f(2) = 2^2 + 1 = 4 + 1 = 5\]Now, add these results together:\[f(x) + f(2) = (x^2 + 1) + 5 = x^2 + 6\]

Key Concepts

Function Evaluation

When we talk about function evaluation, we are looking at how to find the output of a function for a specific input. Given a polynomial function like \(f(x) = x^2 + 1\), we can evaluate it by substituting different values for \(x\). For instance, if we want to find \(f(x+2)\), we replace \(x\) with \(x+2\) in the function. This means every occurrence of \(x\) in the equation \(x^2 + 1\) gets substituted with \(x+2\). Therefore, it becomes

• \(f(x+2) = (x+2)^2 + 1\)

Function evaluation involves these straightforward steps:
- Substitute the input value into the function.
- Simplify the resulting expression if possible.
The primary goal is to determine how the function behaves when different values are plugged in, giving us essential insights into its characteristics.

Expression Simplification

Expression simplification refers to the process of rewriting a mathematical expression in its simplest form. After evaluating functions, like in the case of \(f(x+2)\), the next step is to simplify the expression. We start with \((x+2)^2 + 1\), expand it as follows:

• \((x+2)^2 = x^2 + 4x + 4\)
• Then add the constant: \(x^2 + 4x + 4 + 1\)
• Which further simplifies to \(x^2 + 4x + 5\)

Simplification helps in:
- Making calculations more manageable.
- Reducing complexity of expressions.
It involves combining like terms and performing basic arithmetic operations to arrive at a cleaner, more concise form.

Algebraic Substitution

Algebraic substitution is a fundamental technique in solving equations, where one replaces a variable with a different expression or numerical value. In the given problem, for \(f(x) + f(2)\), we substitute \(x = 2\) into the expression for \(f(x)\). This calculation involves:

• \(f(x) = x^2 + 1\)
• \(f(2) = 2^2 + 1 = 5\)

Now, we add the results from both functions together:

• \(f(x) + f(2) = (x^2 + 1) + 5 = x^2 + 6\)

Algebraic substitution is useful for:
- Solving equations by breaking them down into easier steps.
- Plugging values into formulas to get specific outputs.
This technique simplifies complex algebraic manipulations and aids in problem-solving across various mathematical topics.