Problem 21 - Algebra A Combined Function

Question

Find \((f \circ g)(x)\) and \((g \circ f)(x)\). $$ f(x)=|x| ; g(x)=10 x-3 $$

Step-by-Step Solution

Answer

(f ∘ g)(x) = |10x - 3| and (g ∘ f)(x) = 10|x| - 3.

Step 1: Understand Function Composition

Function composition involves combining two functions, such that the output of one function becomes the input of another function. In this task, we need to find \((f \circ g)(x)\), which means we apply function \(g(x)\) first and then apply function \(f\) to the result. Similarly, for \((g \circ f)(x)\), we apply \(f(x)\) first and then \(g\) to the result.

Step 2: Substitute g(x) into f(x) for (f ∘ g)(x)

Start by calculating \(g(x)\). Given, \(g(x) = 10x - 3\). Now, substitute \(g(x)\) into \(f(x)\): \(f(g(x)) = f(10x - 3)\). Since \(f(x) = |x|\), we have \(f(g(x)) = |10x - 3|\).

Step 3: Simplify (f ∘ g)(x)

Since \(f(10x - 3) = |10x - 3|\), we have the function \((f \circ g)(x) = |10x - 3|\). This expression cannot be simplified further.

Step 4: Substitute f(x) into g(x) for (g ∘ f)(x)

Next, find \((g \circ f)(x)\). Start by calculating \(f(x) = |x|\). Now substitute \(f(x)\) into \(g(x)\): \(g(f(x)) = g(|x|)\). This means substituting \(|x|\) in place of \(x\) in \(g(x) = 10x - 3\), resulting in \(g(|x|) = 10|x| - 3\).

Step 5: Simplify (g ∘ f)(x)

Since \(g(|x|) = 10|x| - 3\), we have the function \((g \circ f)(x) = 10|x| - 3\). This expression also cannot be simplified further.

Key Concepts

Understanding Absolute Value Function

The absolute value function is a fundamental concept in mathematics. It is notated as \(|x|\) and refers to the distance of a number from zero on the number line, regardless of direction. This means:

• For any positive value of \(x\), \(|x| = x\).
• For any negative value of \(x\), \(|x| = -x\).
• For zero, \(|0| = 0\).

This function ensures non-negative output. Its graph creates a 'V' shape on a coordinate plane. In the context of function composition, replacing \(x\) with an expression is pivotal, as seen with \(f(g(x)) = |10x - 3|\). Therefore, absolute value operation applies directly to the result of the expression within, turning any negative result into positive.

Exploring Linear Functions

A linear function is represented by the formula \(g(x) = ax + b\). This is the equation of a straight line and includes:

• Coefficient \(a\), which represents the slope of the line. It determines how steep the line is.
• Constant \(b\), which defines the y-intercept, the point where the line crosses the y-axis.

For the function given, \(g(x) = 10x - 3\):

• The slope is 10, indicating a steep incline.
• The y-intercept is -3.

This straightforward function outputs values directly related to \(x\). When used in function composition, like in \(g(f(x)) = 10|x| - 3\), it processes the output from the previous function with its linear operation. The result is influenced by how \(f(x)\) transforms \(x\).

Function Composition: Combining Functions

Function composition, denoted \((f \circ g)(x)\), involves applying one function to the results of another. It’s like building layers; the output of one becomes the input of the next. Each of these functions transforms the variable in its own way:

• For \((f \circ g)(x) = f(g(x))\), it means applying \(g(x)\) first. Here, \(g(x) = 10x - 3\) is evaluated first to alter \(x\), and then \(f\) is applied, leading to \(|10x - 3|\).
• For \((g \circ f)(x) = g(f(x))\), first, \(f(x) = |x|\) transforms any \(x\) into its absolute value. Then, \(g\) acts on the result: \(10|x| - 3\).

Understanding these processes requires seeing both the transformation each function performs and how those transformations are sequential. This sequence affects the final results which show different operations depending on the order of composition.