Problem 31 - 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

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

Step 1: Understanding the Function

The function given is \( f(x) = x^2 + 1 \). This means for any input \( x \), the function squares the input and then adds 1. We will use this to evaluate the expressions: \( f(x+2) \) and \( f(x) + f(2) \).

Step 2: Evaluating \( f(x+2) \)

Substitute \( x+2 \) for \( x \) in the function \( f(x) = x^2 + 1 \). This gives:\[ f(x+2) = (x+2)^2 + 1 \]Simplify:\[ (x+2)^2 = x^2 + 4x + 4 \]Hence, \[ f(x+2) = x^2 + 4x + 4 + 1 = x^2 + 4x + 5 \]

Step 3: Evaluating \( f(x) + f(2) \)

First, calculate \( f(2) \):\[ f(2) = 2^2 + 1 = 4 + 1 = 5 \]Now use \( f(x) = x^2 + 1 \) and \( f(2) = 5 \) to find \( f(x) + f(2) \):\[ f(x) + f(2) = (x^2 + 1) + 5 = x^2 + 6 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in algebra that consists of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions form the basis for writing equations and functions in mathematics. In this exercise, the expression given is a quadratic one, namely \( f(x) = x^2 + 1 \). Here, **x** is the variable, and the expression represents a rule for calculating outputs from inputs.

• The variable represents an unknown that can hold different values.
• Each part of an expression separated by addition or subtraction is called a term.
• Coefficients are numbers that multiply a variable within a term, like the number 1 within \( x^2 \).

Understanding how to manipulate these expressions is crucial for solving various mathematical problems including evaluating and simplifying functions.

Quadratic Functions

Quadratic functions are specific types of algebraic expressions that take the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These functions graph as parabolas in coordinate planes and have distinct features such as vertices and axes of symmetry. For the function given in our problem, \( f(x) = x^2 + 1 \), it is a simple quadratic function where:

• \( a = 1 \) (leading coefficient)
• \( b = 0 \)
• \( c = 1 \)

Since there is no linear term (\( b = 0 \)), the parabola is symmetric around the y-axis. This simplicity aids in understanding the behavior of the function and evaluating expressions like \( f(x+2) \). Remember:

• The vertex of the parabola is at the point (0, c), in this case, (0,1).
• The minimum value of the function is at this vertex for \( a > 0 \).

Knowing these traits helps explain how the function acts as inputs change.

Function Operations

Function operations involve tasks like addition, subtraction, multiplication, and division of functions. These operations allow us to combine or manipulate functions to form new expressions. In the exercise:

• Evaluating \( f(x+2) \) involved substituting the entire expression \( x+2 \) into the original function. This required expanding \((x+2)^2\) to find the output and simplify it.
• Evaluating \( f(x) + f(2) \) required calculating \( f(2) \) separately, then adding it to \( f(x) \).

When performing function operations, follow these tips:

• Always substitute carefully and simplify step by step.
• Check your calculations to avoid mistakes, especially when dealing with squared terms or complex operations.

These operations allow us to derive new expressions from existing ones and are widely applicable in both pure and applied mathematics.