Problem 27 - Algebra for College Students

Question

If \(f(x)=3 x-2\) and \(g(x)=x^{2}+1\), find \((f \circ g)(-1)\) and \((g \circ f)(3) . \quad 4 ; 50\)

Step-by-Step Solution

Answer

\((f \circ g)(-1) = 4\) and \((g \circ f)(3) = 50\).

Step 1: Understand Composite Functions

The notation \((f \circ g)(x)\) means \(f(g(x))\). Similarly, \((g \circ f)(x)\) means \(g(f(x))\). This involves substituting the output of \(g(x)\) into \(f(x)\) for \((f \circ g)(x)\) or the output of \(f(x)\) into \(g(x)\) for \((g \circ f)(x)\).

Step 2: Find \(g(-1)\) for \((f \circ g)(-1)\)

First, calculate \(g(-1)\) since this will be used in \(f(g(x))\). Given \(g(x) = x^2 + 1\), substituting \(-1\) gives:\[g(-1) = (-1)^2 + 1 = 1 + 1 = 2\]

Step 3: Evaluate \(f(g(-1))\) or \(f(2)\)

Now use the result from Step 2 and substitute it into \(f(x)\). Given \(f(x) = 3x - 2\), compute the value of:\[f(2) = 3(2) - 2 = 6 - 2 = 4\]

Step 4: Find \(f(3)\) for \((g \circ f)(3)\)

Calculate \(f(3)\) since this will be used in \(g(f(x))\). With \(f(x) = 3x - 2\), substitute \(3\) into:\[f(3) = 3(3) - 2 = 9 - 2 = 7\]

Step 5: Evaluate \(g(f(3))\) or \(g(7)\)

Use the result from Step 4 to find \(g(7)\). Given \(g(x) = x^2 + 1\), compute the value of:\[g(7) = 7^2 + 1 = 49 + 1 = 50\]

Key Concepts

Function Composition

When we talk about composite functions, we're referring to the process of combining two or more functions to create a new function. The notation

• \((f \circ g)(x)\) means we first apply function \(g\) and then use its result as the input for function \(f\).
• Similarly, \((g \circ f)(x)\) means we start with function \(f\) and then pass its output to function \(g\).

This is a powerful concept in mathematics, as it allows us to build complex relationships and models by combining simpler functions.
In practical terms, understanding composite functions requires following several steps:

• Calculate the output of the first function.
• Use this output as the input for the second function.
• Combine the results as specified by the problem.

Composite functions are an essential tool, often used in calculus, algebra, and many other areas of mathematics, as they simplify the modeling of complex systems.

Algebraic Functions

Algebraic functions are functions that can be generated using algebraic operations such as addition, subtraction, multiplication, division, and raising to powers. These include polynomial, rational, and radical functions.
For the given problem:

• Function \(f(x) = 3x - 2\) is a linear function, which is a type of polynomial function, characterized by its straight-line graph and a constant rate of change.
• Function \(g(x) = x^2 + 1\) is a quadratic function, another type of polynomial function, known for its parabolic graph and changing rate of growth.

Algebraic functions are foundational in mathematics because they provide the mechanisms to describe real-world phenomena and solve various equations. They give us the tools to manipulate expressions and understand the behavior of different mathematical models.

Problem Solving

When tackling problems involving composite functions, clarity and order in steps are crucial.
To solve composite function problems, follow these practical steps:

Recognize the Problem

Begin by identifying what you're asked to find. In the provided example, we need two composite values:

• \((f \circ g)(-1)\)
• \((g \circ f)(3)\)

Substitute with Care

Use the given functions and substitute as necessary:

• For \((f \circ g)(-1)\):
• Compute \(g(-1)\), and use this result to find \(f(g(-1))\).
• This involves calculating via the algebraic rule \(f(x)\).
• For \((g \circ f)(3)\):
• First, find \(f(3)\), then use it for \(g(f(3))\).

Check Your Calculations

Double-check each substitution and numerical computation to ensure accuracy.
By breaking down actions into these straightforward steps, solving composite function exercises becomes more systematic and less daunting. Remember, practice and these clear strategies improve both understanding and skill.