Problem 19 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=x+4, g(x)=2 x+1$$

Step-by-Step Solution

Answer

a. \((f \circ g)(x) = 2x + 5\) b. \((g \circ f)(x) = 2x + 9\) c. \((f \circ g)(2) = 9\)

Step 1: Solving for \((f \circ g)(x)\)

The \((f \circ g)(x)\) is read as 'f of g of x', meaning, replace every instance of 'x' in function f, with the entire function g.Using the given functions \(f(x) = x + 4\) and \(g(x) = 2x + 1\), we substitute \(g(x)\) into \(f(x)\). Resulting in: \((f \circ g)(x) = (2x + 1) + 4 = 2x + 5\)

Step 2: Solving for \((g \circ f)(x)\)

The \((g \circ f)(x)\) is read as 'g of f of x', meaning we substitute function \(f(x)\) into function \(g(x)\). Using the given functions \(f(x) = x + 4\) and \(g(x) = 2x + 1\), substitute \(f(x)\) into \(g(x)\) resulting in: \((g \circ f)(x) = 2(x + 4) + 1 = 2x + 8 + 1 = 2x + 9\)

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

To find \((f \circ g)(2)\), which is read as 'f of g at 2', we need to substitute '2' into the function we found for \((f \circ g)(x)\), which was \(2x + 5\). So, \((f \circ g)(2) = 2*2 + 5 = 4 + 5 = 9\)

Key Concepts

Composite Functions

Composite functions are an essential concept in mathematics, particularly in algebra and calculus. They involve the combination of two or more functions, where the output of one function becomes the input for another. The notation

• \((f \circ g)(x)\) is read as "f of g of x," meaning that the function \(g(x)\) is evaluated first, and then its output is used as the input for \(f(x)\).
• For \((g \circ f)(x)\), the function \(f(x)\) is evaluated first, followed by \(g(x)\).

This concept is used to solve complex problems by breaking them down into simpler steps. By understanding each function individually and then combining them, new insights and solutions can be reached. It's a technique that requires clear understanding and precision, as reversing the order of functions changes the result.

Function Evaluation

Function evaluation is a fundamental process in mathematics where we find the output of a function for a specific input. To evaluate a function, follow these steps:

• Identify the given function. In this case, we have \(f(x) = x + 4\) and \(g(x) = 2x + 1\).
• Substitute the given value into the function. If you're evaluating \((f \circ g)(2)\), you first compute \(g(2)\) to find the input for \(f\).
• Compute the result. For instance, \(g(2) = 2 \times 2 + 1 = 5\), and then \(f(5) = 5 + 4 = 9\).

This process helps find the output for specific inputs and can be extended to evaluate composite functions by ensuring each function in the composition is evaluated in the correct order.

Algebraic Substitution

Algebraic substitution is a critical skill that involves replacing variables in a mathematical expression or function with their corresponding expressions or values. This is an essential technique used in solving composite functions.When you're solving

• \((f \circ g)(x)\), you substitute \(g(x)=2x+1\) into \(f(x)=x+4\), resulting in \(f(g(x)) = 2x + 1 + 4 = 2x + 5\).
• For \((g \circ f)(x)\), substitute \(f(x)=x+4\) into \(g(x)=2x+1\), resulting in \(g(f(x)) = 2(x+4) + 1 = 2x + 9\).

Remember, the correct substitution of expressions is crucial. Missteps in calculating substitutions can lead to incorrect results. Accurate and careful substitution is a part of what makes problem-solving in math both challenging and rewarding.