Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Let \(f(x)=x^{2}\) and \(g(x)=2 x-5 .\) Match each function in Group I with the correct expression in Group $I$$$I$$$$(f \circ g)(x)$$$$\mathbf{II}$$$$A.\quad4 x^{2}-20 x+25$$$$B.\quad x^{2}-2 x+5$$$$C.\quad 2 x^{2}-5$$$$D.\quad \frac{x^{2}}{2 x-5}$$$$E. \quad x^{2}+2 x-5$$$$F. \quad 2 x^{3}-5 x^{2}$$

Step-by-Step Solution

Answer

(f \circ g)(x) = 4x^2 - 20x + 25, which matches with A.

Step 1: Understand the Composition of Functions

The composition of functions \((f \circ g)(x)\) means we need to substitute the function \(g(x)\) into the function \(f(x)\). In mathematical terms, this means finding \(f(g(x))\).

Step 2: Find g(x)

Begin by recalling that \(g(x) = 2x - 5\). This expression will be substituted into \(f(x)\) to find \(f(g(x))\).

Step 3: Substitute g(x) into f(x)

Since \(f(x) = x^2\), we substitute \(g(x)\) into \(f(x)\). This gives us \(f(g(x)) = (g(x))^2 = (2x - 5)^2\).

Step 4: Expand (2x - 5)^2

To expand \((2x - 5)^2\), use the binomial expansion formula: \((a - b)^2 = a^2 - 2ab + b^2\). Apply this to the terms: \((2x)^2 - 2(2x)(5) + 5^2\).

Step 5: Simplify the Expression

Calculate each part: \((2x)^2 = 4x^2\), \(-2(2x)(5) = -20x\), \(5^2 = 25\). Combine these terms to obtain: \(4x^2 - 20x + 25\).

Step 6: Match with Group II

Compare the simplified expression \(4x^2 - 20x + 25\) with the options provided in Group II. Look for the exact match.

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. In the given exercise, you deal with a quadratic polynomial function, defined as \( f(x) = x^2 \). Polynomial functions are widely used due to their simplicity and the ability to model various phenomena in fields like physics, engineering, and finance.
When analyzing a polynomial function, you should note:

• The degree of the polynomial, which is the highest power of the variable present (e.g., 2 for \(x^2\)).
• Coefficients, which are the numbers multiplying the terms (e.g., 4 in \(4x^2\)).
• The shape of its graph, typically forming curves such as parabolas, hyperbolas, etc., depending on its degree.

Grasping the structure of polynomial functions is fundamental for manipulating equations and finding solutions.

Binomial Expansion

Binomial expansion involves expanding expressions raised to a power, like \((2x - 5)^2\), using known formulas such as \((a - b)^2 = a^2 - 2ab + b^2\). This technique provides a systematic approach to simplifying expressions and calculating powers without manually multiplying each term.
The process generally requires you to:

• Identify the 'a' and 'b' values in the binomial.
• Apply the expansion formula accurately.
• Simplify the results to form a polynomial.

In our exercise, \( (2x - 5)^2 \) becomes \(4x^2 - 20x + 25\), showcasing the expanded form. Mastering binomial expansions is crucial for algebraic manipulation and helps in solving higher-degree polynomial equations efficiently.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions. This includes operations like expansion, factorization, addition, subtraction, and simplification of expressions. In our exercise, expanding \((2x - 5)^2\) involves several algebraic steps to reach a simplified polynomial form.
To excel at algebraic manipulation, you should:

• Become familiar with different algebraic identities, such as binomial theorems.
• Develop skills in combining like terms efficiently.
• Practice consistently to recognize patterns and simplify complex expressions.

Strong algebraic manipulation skills are foundational for more complex areas of mathematics, enabling you to solve a wide array of equations and mathematical problems.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as +, −, ×, ÷) that represent a specific value or function. In the exercise, you encounter multiple expressions, such as \(g(x) = 2x - 5\) and \(f(x) = x^2\), which are combined through function composition.
These expressions can involve:

• Variables (e.g., \(x\)), which act as placeholders for unknown values.
• Operations, signifying processes like addition or multiplication.
• Constants, which are fixed numeric values within the expression.

Understanding how to interpret and manipulate mathematical expressions is vital as it lays the groundwork for solving equations, modeling situations, and exploring mathematical structures effectively.