Problem 1 - Algebra 2 and Trigonometry

Question

Marcie said that if \(f(x)=x^{2},\) then \(f(a+1)=(a+1)^{2} .\) Do you agree with Marcie? Explain why or why not.

Step-by-Step Solution

Answer

Yes, Marcie is correct; \( f(a+1) = (a+1)^{2} = a^{2} + 2a + 1 \).

Step 1: Define the Problem

We need to verify if the function transformation provided by Marcie is correct. Specifically, we need to evaluate the function \( f(x) = x^{2} \) for \( x = a+1 \) and see if it results in \( (a+1)^{2} \).

Step 2: Substitute Into the Function

The expression \( f(a+1) \) asks us to substitute \( a+1 \) into the function \( f(x) = x^{2} \). Therefore, we replace \( x \) with \( a+1 \) to get \( f(a+1) = (a+1)^{2} \).

Step 3: Expand the Expression

To understand the expression \((a+1)^{2}\), we expand it using the binomial theorem. This gives us \((a+1)^{2} = a^{2} + 2a + 1\).

Step 4: Compare Results

The expanded form \( a^{2} + 2a + 1 \) matches the definition of the function \( f(x) = x^{2} \) applied to \( x = a+1 \), confirming that \( f(a+1) = (a+1)^{2} \) is indeed correct.

Key Concepts

Function Evaluation

Function evaluation is an essential concept in algebra that involves determining the value of a function at a specific point. Functions like \(f(x) = x^2\) define a relationship between the input \(x\) and the output \(f(x)\). When you evaluate a function, you replace the variable \(x\) with a specific value or expression to find the outcome.

In the exercise at hand, Marcie evaluates the function \(f(x) = x^2\) at \(x = a+1\). Imagine having a function as a machine that processes inputs. For example, if your machine squares numbers, you simply insert a number or an expression, then observe what comes out.

This substitution process involves replacing \(x\) with \(a+1\) in the function, transforming it into \(f(a+1) = (a+1)^2\). Understanding this helps in situations where you need to predict the behavior of functions under different conditions.

• Recognize what a function does and how it transforms inputs.
• Learn to substitute precisely and evaluate functions confidently.

Binomial Theorem

The binomial theorem is a powerful tool in algebra for expanding expressions that are raised to a power. It allows you to systematically expand expressions like \((a+b)^n\), where \(n\) is a natural number.

In the context of our exercise, the expression \((a+1)^2\) needs to be expanded. By applying the binomial theorem, this can be expanded into \(a^2 + 2a + 1\). This expansion follows the formula:
\[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
Where \( \binom{n}{k} \) is a binomial coefficient, which represents the number of ways to pick \(k\) outcomes from \(n\) possibilities.

• The binomial theorem simplifies the process of multiplying binomials over several terms.
• It helps in computing powers of sums and understanding polynomial behavior.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, and multiplication. They are the building blocks in algebra and are used to represent mathematical relationships.

In the step-by-step solution, when we expand \((a+1)^2\) to \(a^2 + 2a + 1\), we're working with algebraic expressions. These expressions must be manipulated according to algebraic rules to simplify or to be set up for further operations.

This clearly shows how components are grouped and rearranged, making complex problems more approachable. We often need to combine like terms, distribute numbers, or factorize expressions in algebra.

• Understand different algebraic operations and their hierarchies.
• Learn to manipulate expressions to solve equations and simplify forms.
• Algebraic expressions lay the groundwork for solving real-world problems.