Problem 87 - Algebra 2

Question

Let \(f(x)=2 x-3\) and \(g(x)=-x^{2}-1 .\) Find \((g \circ f)(x)\) $$\begin{array}{ll}{\text { A. }-2 x^{3}+3 x^{2}-2 x+3} & {\text { B. }-4 x^{2}+12 x-10} \\ {\text { C. }-x^{2}+2 x-4} & {\text { D. }-x^{2}-2 x+2}\end{array}$$

Step-by-Step Solution

Answer

The composition of the functions \(f(x)\) and \(g(x)\), \((g \circ f)(x)\), is \(-4x^2 + 12x - 10\) which corresponds to answer B.

Step 1: Understanding Function Composition

Function composition \((g \circ f)(x)\) means substituting \(f(x)\) into \(g(x)\). This means for every 'x' in \(g(x)\), we will substitute with the function \(f(x)\).

Step 2: Substituting \(f(x)\) into \(g(x)\)

Now let's substitute \(f(x) = 2x - 3\) into \(g(x)\). So wherever we see 'x' in \(g(x) = -x^2 - 1\), we replace it with \(2x - 3\). So the composition \((g \circ f)(x)\) is: \(g(f(x)) = -(2x - 3)^2 - 1\).

Step 3: Simplifying the Expression

Let's simplify the expression further. Remember to use the square expansion formula: \((a - b)^2 = a^2 - 2ab + b^2\). So the function \(-(2x - 3)^2 - 1\) simplifies to \(-4x^2 + 12x - 9 - 1 = -4x^2 + 12x - 10\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, linked together by addition, subtraction, and multiplication operations. Each part of the polynomial expression is called a term. Terms are formed by multiplying a constant (called a coefficient) by one or more variables raised to whole number powers. Some key aspects of polynomials include:

• The degree of the polynomial, which is determined by the highest power of the variable(s) in any term.
• The leading term, which is the term with the highest degree, and its coefficient is known as the leading coefficient.
• Polynomials can be classified based on their degree: linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.

For instance, the polynomial \(-4x^2 + 12x - 10\) is a quadratic polynomial because the highest power of the variable \(x\) is 2. Understanding these components is vital as they are often the foundation for more complex algebraic problems, such as function compositions.

Quadratic Functions

A quadratic function is a specific type of polynomial that features a degree of 2. It generally takes the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Quadratic functions are recognized by their characteristic parabolic graphs, which can either open up or down, depending on the sign of \(a\).In the context of function composition, when we substitute one function into another, we often end up with a quadratic function. This happens because the substitution involves squaring a linear expression, resulting in components that align with the form of a quadratic.Key features of quadratic functions include:

• The vertex, which is the highest or lowest point of the parabola.
• The axis of symmetry, a vertical line that passes through the vertex, splitting the parabola into two mirror-image halves.
• The roots or zeros, which are the x-values where the function equals zero.

Understanding quadratic functions helps in visualizing and solving real-world problems, as well as simplifying complex algebraic expressions like the one made from the composite function \(-(2x - 3)^2 - 1\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions are fundamental to algebra and serve as the building blocks for forming equations and functions.A key skill in working with algebraic expressions is the ability to simplify them efficiently. Simplifying involves combining like terms (terms with the same variable and exponent) and using distributive, associative, and commutative properties of operations.When dealing with function compositions, such as \(g(f(x)) = -(2x - 3)^2 - 1\), the simplifying process is about applying these properties to break down the expression into a simpler form like \(-4x^2 + 12x - 10\). Doing so not only makes the expression easier to work with but also reveals insights about its behavior, such as the degree of the resulting polynomial and its effect on graphing.Mastering algebraic expressions gives students the tools to tackle a wide range of mathematical challenges, enabling them to manipulate and evaluate expressions accurately and efficiently.