Problem 59 - College Algebra Essentials

Question

Find a. \((f \circ g)(x) \quad \) b. \((g \circ f)(x) \quad \) c. \((f \circ g)(2) \quad \) d. \((g \circ f)(2)\) $$f(x)=4-x, g(x)=2 x^{2}+x+5$$

Step-by-Step Solution

Answer

The function compositions are a. \((f \circ g)(x) = -2x^{2} - x -1\), b. \((g \circ f)(x) = 2x^{2} - 16x + 21\), c. \((f \circ g)(2) = -9\) and d. \((g \circ f)(2) = -7\).

Step 1: Find \((f \circ g)(x)\)

Substitute \(g(x)\) into \(f(x)\) wherever there is \(x\). So \(f(g(x)) = f(2x^{2}+x+5) = 4 - (2x^{2}+x+5)\). Simplifying this gives \(f(g(x)) = -2x^2 - x -1 .

Step 2: Find \((g \circ f)(x)\)

Substitute \(f(x)\) into \(g(x)\) wherever there is \(x\). So \(g(f(x)) = g(4-x) = 2(4-x)^{2}+(4-x)+5\). Simplifying this gives \(g(f(x)) = 2x^{2} - 16x + 21.

Step 3: Find \((f \circ g)(2)\)

Substitute \(x = 2\) into the function composition found in Step 1. So \(f(g(2)) = -2(2)^{2} - 2 -1 = -9.\)

Step 4: Find \((g \circ f)(2)\)

Substitute \(x = 2\) into the function composition found in Step 2. So \(g(f(2)) = 2(2)^{2} - 16(2) + 21 = -7\).

Key Concepts

Composition of Functions

The composition of functions is a fundamental concept in algebra that deals with combining two functions into a single function. This process is symbolized by \(f \circ g\), which means 'the function \(f\) composed with \(g\).' To perform function composition, we substitute the output of the second function \(g(x)\) directly into the first function \(f(x)\). For example, if \(f(x) = 4 - x\) and \(g(x) = 2x^2 + x + 5\), then the composition \(f \circ g\) is defined as \(f(g(x)) = f(2x^2 + x + 5)\). Comprehending this concept is crucial for understanding how functions interact with each other.

When approaching function composition exercises, remember:

• Function order matters; \(f \circ g\) is not the same as \(g \circ f\).
• Always start by substituting the inside function into the outside one.
• Simplify the resulting function as much as possible.

Algebraic Functions

Algebraic functions are expressions that use algebraic operations, such as addition, subtraction, multiplication, division, and exponentiation, on variables. The functions \(f(x) = 4 - x\) and \(g(x) = 2x^2 + x + 5\) from our exercise are both algebraic functions. These functions are the building blocks of algebra and are essential for modeling real-world situations.

When working with algebraic functions:

• Pay attention to the domain, as some operations may not be defined for all real numbers.
• Manipulating algebraic functions involves applying the distributive property, combining like terms, and factoring when possible.
• Understanding how to manipulate these functions enables you to work with more complex algebraic expressions and function compositions.

Function Evaluation

Evaluating a function means finding the value of the function for a specific value of \(x\). This is a significant skill in algebra, as it allows us to understand the behavior of functions at particular points. To evaluate a function, we replace \(x\) with the given number and simplify the expression. For instance, to find \(f \circ g)(2)\), we substitute \(x = 2\) into the composition \(f(g(x))\) that we calculated earlier.

Remember the following tips when evaluating functions:

• Replace the variable with the given value accurately.
• Simplify the expression thoroughly to arrive at the final answer.
• Evaluation is a tool that helps in graphing functions and solving equations.

College Algebra

College algebra is an advanced course that delves into various algebraic concepts, functions, and their applications. It often marks a student's first foray into higher-level mathematics. Grasping core concepts such as function composition, algebraic functions, and function evaluation is vital. These topics provide a foundation for later studies in calculus, statistics, and other mathematical fields.

In college algebra, students learn to:

• Analyze and solve complex algebraic equations.
• Work with different types of functions, including linear, polynomial, rational, and exponential.
• Understand the theoretical aspects of algebra and how they apply to real-world problems.

Developing proficiency in these areas is paramount for any student looking to excel in mathematics or related disciplines.