Problem 18 - College Algebra

Question

Find: a. \((f \circ g)(x)\) b. \(\left(g^{\circ} f\right)(x)\) c. \((f \circ g)(2)\) $$f(x)=3 x, g(x)=x-5$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = 3x - 15\), \((g \circ f)(x) = 3x - 5\) and \((f \circ g)(2) = -9\)

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

First, we calculate \((f \circ g)(x)\) by substituting \(g(x)\) into the function \(f(x)\). We have: \[f(g(x)) = f(x - 5)\] By direct substitution we get: \[f(g(x)) = 3(x - 5)\] So, \((f \circ g)(x) = 3x - 15\].

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

Next is to calculate \((g \circ f)(x)\) by substituting \(f(x)\) into function \(g(x)\). We have: \[g(f(x)) = g(3x)\] By direct substitution we get: \[g(f(x)) = 3x - 5\] So, \((g \circ f)(x) = 3x - 5\].

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

Lastly, we evaluate \((f \circ g)(2)\) by substituting \(x = 2\) into \((f \circ g)(x)\). From step 1, we know that \((f \circ g)(x) = 3x - 15\). Therefore, \((f \circ g)(2)= 3*2 - 15 = 6 - 15 = -9\]

Key Concepts

Composite Functions

A composite function is formed when one function is applied immediately after another function. This operation is denoted by the symbol \(\circ\) and is understood as 'f composed with g', written as \(f \circ g\). It's crucial to interpret this correctly: the result of applying function g to an input x is used as the input for function f.

For example, in the provided exercise, \(f(x) = 3x\) and \(g(x) = x - 5\), the composite \(f \circ g\) is computed by first taking any value of x, substituting it into g to find \(g(x)\), and then substituting \(g(x)\) into f. This results in \(f(g(x)) = 3(x - 5)\), or more simply, \(3x - 15\).

To visualize the process, imagine a flow where x first flows through g, becoming \(g(x)\), and then this outcome flows through f, resulting in \(f \circ g\) of x. The order is critical as \(f \circ g\) can yield a different result compared to \(g \circ f\), hence function composition is not generally commutative.

Function Operations

In mathematics, function operations such as addition, subtraction, multiplication, and division can be performed on functions in the same manner as numbers. However, when we talk about operations on functions, there’s a unique operation, called function composition, which isn’t as straightforward as the others.

Coming back to composition, understand it as an 'operation within an operation'. We first perform g, and then we perform f on the result of g. Our exercise used two functions, f and g, to demonstrate function composition. If we were to perform other operations such as addition or multiplication, the processes might look like \(f(x) + g(x)\) or \(f(x) \cdot g(x)\), respectively.

It is crucial to master these operations because they provide a foundational skill for understanding more complex mathematical topics. These operations also maintain many properties similar to regular arithmetic, but with essential differences such as non-commutativity, as seen with function composition.

Function Evaluation

The process of function evaluation is like substituting a specific value for the variable of the function and calculating the corresponding output. Let's apply what we learned from the composition of functions to the evaluation step.

Consider \(f \circ g\) from our previous discussion, where we calculated \(f \circ g\)(x) to be \(3x - 15\). To evaluate this composite at x = 2, denoted as \(f \circ g\)(2), we simply substitute 2 in place of x which gives us \(3 \cdot 2 - 15\). The result is -9. Hence, function evaluation at a particular point involves substituting and performing the arithmetic operations as per the function’s rule.

Always remember to perform the evaluation step by following the function's operations in the correct order. Every step of calculation must be precise to ensure the accuracy of the final result, as a small error in evaluation can lead to a completely incorrect output.