Problem 64 - College Algebra Essentials

Question

Find a. \((f \circ g)(x) \quad \) b. \((g \circ f)(x) \quad \) c. \((f \circ g)(2) \quad \) d. \((g \circ f)(2)\) $$f(x)=6 x-3, g(x)=\frac{x+3}{6}$$

Step-by-Step Solution

Answer

The solutions for the exercise are \((f\circ g)(x)=x\), \((g\circ f)(x)=x\), \((f\circ g)(2)=2\) and \((g\circ f)(2)=2\)

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

To find \((f\circ g)(x)\), substitute \(g(x)\) into \(f(x)\). Therefore, \((f\circ g)(x)\) becomes \(f(g(x))\). Given \(f(x)=6x-3\) and \(g(x)=(x+3)/6\), substituting \(g(x)\) into \(f(x)\), it becomes \(f(g(x))=6(g(x))-3=6((x+3)/6)-3\) after simplification, \((f\circ g)(x)=x+3-3=x\).

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

To find \((g\circ f)(x)\), we substitute \(f(x)\) into \(g(x)\). Therefore, \((g\circ f)(x)\) becomes \(g(f(x))\). Given \(f(x)=6x-3\) and \(g(x)= (x+3)/6\), replace \(f(x)\) in \(g(x)\), it becomes \(g(f(x)) = (f(x)+3)/6=((6x-3)+3)/6\) after simplifying, \((g\circ f)(x)=x\).

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

To find \((f\circ g)(2)\), substitute 2 into the function \((f\circ g)(x)\) obtained in Step 1. After substituting 2 into \((f\circ g)(x)\), it becomes \((f\circ g)(2)=2\).

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

To find \((g\circ f)(2)\), substitute 2 into the function \((g\circ f)(x)\) obtained in step 2. So after substituting 2 into \((g\circ f)(x)\), it becomes \((g\circ f)(2)=2\).

Key Concepts

Step-by-Step Solution

When tackling problems in mathematics, especially those involving function composition, taking a step-by-step approach is crucial. Let's break down the process:

• Start by identifying the functions involved. In our problem, we have two functions, \(f(x) = 6x - 3\) and \(g(x) = \frac{x+3}{6}\).
• Next, determine the composition of the functions, i.e., \((f \circ g)(x)\) and \((g \circ f)(x)\). This involves substituting one function into the other.
• Calculate the specific values for these compositions when \(x = 2\). This requires you to substitute the value into the composite functions you've obtained.

Understanding each step clearly and performing substitutions correctly are key to solving these problems successfully. This not only provides the correct answer but also builds a strong foundation for understanding function compositions.

Algebraic Functions

Algebraic functions are the backbone of many mathematical processes. Here, they primarily involve operations such as addition, subtraction, multiplication, and division. The given functions for our exercise, \(f(x) = 6x - 3\) and \(g(x) = \frac{x+3}{6}\), are both algebraic functions.

• \(f(x) = 6x - 3\) is a linear function. It involves multiplication of \(x\) by 6 and then subtraction of 3.
• \(g(x) = \frac{x+3}{6}\) is a rational function because it includes division. The expression \(x + 3\) is divided by 6.

Combing these functions through composition is an important technique. It allows for new and interesting functions and can simplify complex problems by seeking patterns in the compositions.

Function Evaluation

Function evaluation is a fundamental notion where a specific input value is substituted into a function to yield an output. This skill is essential when working with function compositions such as \((f \circ g)(x)\) or \((g \circ f)(x)\).

• To evaluate \((f \circ g)(2)\), start with \((f \circ g)(x) = x\). Substitute 2 for \(x\) to find \((f \circ g)(2) = 2\).
• For \((g \circ f)(2)\), use the composition \((g \circ f)(x) = x\). Again, substitute 2 to obtain \((g \circ f)(2) = 2\).

This method demonstrates that, even with complex-looking compositions, the evaluation can sometimes lead to straightforward solutions. Understanding and practicing these substitutions strengthen mathematical fluency and application skills.