Problem 8 - 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. $$(g \circ f)(-2)$$

Step-by-Step Solution

Answer

The result is -5.

Step 1: Understand the Composition Notation

The notation \((g \circ f)(x)\) means that you first apply function \(f(x)\) and then use the result as the input for function \(g(x)\). This is the same as saying \(g(f(x))\).

Step 2: Calculate \(f(-2)\)

First, substitute \(-2\) into the function \(f(x) = x^2 + 3x\):\[f(-2) = (-2)^2 + 3(-2) = 4 - 6 = -2\].

Step 3: Substitute the Result into \(g(x)\)

Take the result from Step 2, which is \(f(-2) = -2\), and substitute it into the \(g(x)\) function:\[g(-2) = 2(-2) - 1 = -4 - 1 = -5\].

Step 4: Conclude the Composition Result

Therefore, the result of the composition \((g \circ f)(-2)\) is \(-5\).

Key Concepts

Polynomial Functions

Polynomial functions are a key foundation in algebra and mathematics. They are expressions consisting of variables and coefficients, structured in terms of powers of the variables. For example, in the polynomial function given in our problem, \( f(x) = x^2 + 3x \):

• The terms are \( x^2 \) and \( 3x \).
• The highest exponent of the variable is 2, indicating that it is a quadratic polynomial.
• Quadratic polynomials graph as parabolas, which are key in modeling various real-world phenomena such as projectile motion.
• Polynomial functions can have multiple terms, consisting of both positive and negative exponents.

Understanding these features allows us to recognize how polynomials fit into the broader spectrum of mathematical functions and their applications.

Algebraic Operations

Algebraic operations on functions involve implementing basic operations like addition, subtraction, multiplication, and composition. These operations allow us to transform and manipulate functions in order to solve problems or simplify expressions. In the composition

• The operation we perform is called 'function composition'.
• It's symbolized as \((g \circ f)(x)\), meaning apply \( f(x) \) first and \( g(x) \) second.
• For our exercise, it's about taking the output of \( f(x) \) and plugging it into \( g(x) \).
• This step-by-step process ensures accuracy and a clear understanding of the interaction between different functions.

Mastering these operations helps in tackling more complex algebraic expressions and deriving solutions efficiently.

Step-by-Step Problem Solving

Step-by-step problem solving is an essential skill that allows you to break down complex problems into manageable steps. Let's see how it works with a function composition problem:

• Start by understanding the notation, which tells you the order of operations.
• Identify what each function represents. In our case, \( f(x) \) and \( g(x) \).
• Calculate the first function with the given value, such as \( f(-2) \) in the exercise. This involves substituting the value into the function and simplifying.
• Take the result from the first calculation and use it in the second function, \( g(x) \), to find the final solution.
• This structured approach not only confirms you're executing operations correctly but also builds confidence when faced with similar mathematical challenges.

By following these simple steps, you enhance your critical thinking skills and improve your ability to solve mathematics problems methodically and accurately.