Problem 9 - A Graphical Approach to College Algebra

Question

Let \(f(x)=x^{2}+3 x\) and \(g(x)=2 x-1 .\) Perform the composition or operation indicated. $$(f \circ g)(x)$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 4x^2 + 2x - 2\).

Step 1: Understand Function Composition

The composition of two functions \(f\) and \(g\), denoted by \((f \circ g)(x)\), means that you first apply the function \(g(x)\) and then apply the function \(f(x)\) to the result. Mathematically, it is represented as \(f(g(x))\).

Step 2: Substitute g(x) into f(x)

Start by substituting \(g(x) = 2x - 1\) into \(f(x)\). The function \(f(x)\) is given by \(f(x) = x^2 + 3x\). So, we need to find \(f(g(x)) = f(2x-1)\).

Step 3: Apply f to g(x)

Substitute \(2x - 1\) everywhere you see \(x\) in \(f(x) = x^2 + 3x\). This gives us: \[ f(2x-1) = (2x-1)^2 + 3(2x-1) \] Expand and simplify this expression to compute \(f(g(x))\).

Step 4: Expand \((2x-1)^2\)

Calculate \((2x-1)^2\): \[ (2x-1)^2 = (2x-1)(2x-1) = 4x^2 - 4x + 1 \] This is the expansion of \((2x-1)^2\).

Step 5: Simplify f(g(x))

Now, include the result of \((2x-1)^2\) in the expression: \[ f(2x-1) = 4x^2 - 4x + 1 + 3(2x-1) \]Distribute the 3 in \(3(2x-1)\), which gives: \[ 3(2x-1) = 6x - 3 \] Combine all the terms: \[ f(2x-1) = 4x^2 - 4x + 1 + 6x - 3 \] Simplify by combining like terms:

Step 6: Final Answer

Combine the like terms: \[ f(2x-1) = 4x^2 + 2x - 2 \] Thus, \((f \circ g)(x) = 4x^2 + 2x - 2\).

Key Concepts

Polynomial Functions

Polynomial functions are vital building blocks in algebra and calculus. They are mathematical expressions consisting of variables and coefficients, constructed using only addition, subtraction, multiplication, and non-negative integer exponents of variables.

A general polynomial function is expressed as:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_2x^2 + a_1x + a_0 \)

Here, \( a_n, a_{n-1}, \ldots , a_0 \) are constants known as coefficients, and \( n \) is a non-negative integer named the degree of the polynomial.

The degree of the polynomial typically determines its general shape and behavior as \( x \) approaches infinity or negative infinity. In our original exercise, both \( f(x) = x^2 + 3x \) and \( g(x) = 2x - 1 \) are polynomials. Specifically, \( f(x) \) is a quadratic polynomial function of degree 2, as it includes an \( x^2 \) term, and \( g(x) \) is a linear polynomial function of degree 1, as it involves only \( x \).

Understanding polynomial functions is crucial because they often serve as approximations for more complex functions in various mathematical analyses.

Algebraic Operations

Algebraic operations refer to basic mathematical procedures applied to manipulate algebraic expressions. They include operations such as addition, subtraction, multiplication, and division of algebraic expressions. These operations form the foundation for solving many algebraic equations and understanding more complex mathematical concepts.

In the context of our exercise, algebraic operations play a significant role:

• Substitution: In function composition, we substitute one function into another. For our functions \( f(x) \) and \( g(x) \), we substituted \( g(x) = 2x - 1 \) into \( f(x) \) by replacing \( x \) in \( f(x) = x^2 + 3x \) with \( 2x - 1 \).
• Expansion: This involves multiplying expressions to open out brackets, such as expanding \((2x - 1)^2\). In this exercise, multiplications are carried out to simplify the expression \( (2x - 1)(2x - 1) \) to obtain \( 4x^2 - 4x + 1 \).
• Simplification: After expanding and combining terms, we simplify expressions by combining like terms. This operation ensures that our answer is presented in the simplest form, like reducing \( 4x^2 - 4x + 1 + 6x - 3 \) to \( 4x^2 + 2x - 2 \).

Algebraic operations are essential tools that allow us to distill complex expressions into more manageable forms for further analysis or application.

Quadratic Expressions

Quadratic expressions are a subset of polynomial expressions characterized primarily by the degree of 2. A general quadratic expression has the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constant coefficients, and \( a eq 0 \).

The expression \( f(x) = x^2 + 3x \) is quadratic because it contains an \( x^2 \) term. Quadratic expressions are remarkable for the parabolic shapes they form when graphed. These parabolas can open upwards or downwards, depending on the sign of \( a \). When \( a > 0 \), the parabola opens upwards, and when \( a < 0 \), it opens downwards.

In our problem, the quadratic expression arises again in our final composed function \( f(g(x)) = 4x^2 + 2x - 2 \). When we perform the composition, the quadratic nature is preserved, resulting in another quadratic expression.

• Root finding: This is the process of determining where the quadratic expression equals zero, commonly done through techniques such as factoring, completing the square, or using the quadratic formula.
• Vertex form: Quadratic expressions can often be rewritten to identify their vertex, allowing for easy graphing. This is done through completing the square for expressions like \( 4x^2 + 2x - 2 \).
• Application: Quadratics are ubiquitous in real-world scenarios such as projectile motion, where the height of an object as a function of time follows a parabolic path.

Understanding quadratic expressions enriches our ability to explore mathematical functions and their real-world applications with more precision and insight.