Problem 78 - Algebra 2

Question

Let \(f(x)=4 x, g(x)=\frac{1}{2} x+7,\) and \(h(x)=|-2 x+4| .\) Simplify each function. $$ (h \circ(g \circ f))(x) $$

Step-by-Step Solution

Answer

The simplified form of the composition function \(h \circ (g \circ f)(x) = |-4x -10|\) splits into two separate functions for different ranges of \(x\): \(4x + 10\) when \(x \geq 2.5\) and \(-4x - 10\) when \(x < 2.5\).

Step 1: Compute 'g' composed with 'f'

First, compose function \(g\) with function \(f\). To do so, substitute \(f(x)\) in place of \(x\) in \(g(x)\). Hence,\(g(f(x))=g(4x)=(1/2)*(4x)+7 = 2x + 7.

Step 2: Compute 'h' composed with 'g composed with f'

Now, substitute the result from step 1 (which is \(g \circ f\)) into function \(h\). This lets us find \(h(g(f(x)))\). So we have,\(h(g(f(x)))=h(2x + 7)= |(-2)*(2x+7) + 4| = |-4x -14 + 4| = |-4x -10|.

Step 3: Simplify Absolute Value Function

The absolute value function is already simplified. To further simplify it, we can split the absolute function into two separate cases: \nWhen \(x \geq 2.5\), \(-4x -10\) is negative. Taking the absolute value, we obtain the function \(4x + 10\).\nWhen \(x < 2.5\), \(-4x -10\) is positive or zero. Taking the absolute value, we retain the same function. Therefore, \n\(h \circ (g \circ f)(x) = |-4x -10| = \begin{cases} 4x + 10, & x \geq 2.5 \ -4x - 10, & x < 2.5\end{cases}\)

Key Concepts

Piecewise Functions

Whether you are just beginning to navigate functions or you're quickly advancing, piecewise functions are not something to overlook. These are functions defined by multiple sub-functions, where each sub-function is relevant in a particular interval of the domain. Simply put, a piecewise function can have different rules depending on the input value.
If you consider the function given in the exercise \[ h \circ (g \circ f)(x) = |-4x -10| \], you'll notice that it transforms into a piecewise function as we simplify it further. Here is how it divides:

• For \(x \geq 2.5\), the expression \(-4x -10\) inside the absolute value is negative, so we change the sign when taking the absolute value.
• For \(x < 2.5\), it remains the same because the expression is non-positive.

This results in:\[h \circ (g \circ f)(x) = \begin{cases} 4x + 10, & x \geq 2.5 \ -4x - 10, & x < 2.5 \end{cases}\] Piecewise functions are practical for modeling scenarios where a function behaves differently over different segments of its domain.

Absolute Value

The absolute value function is one of those mathematical operations that raises curiosity because of its unique property—it makes anything inside of it non-negative. It measures the "distance" of a number from zero on the number line, regardless of direction.
In the given exercise, the absolute value function appears as \(|-4x -10|\). The goal is to eliminate any negative values while simplifying.
In general, if inside the absolute value, the expression \(a\) is non-positive, the absolute value \(|a|\) will flip the sign to make it positive. Conversely, if it is already positive or zero, it remains unchanged.

• This makes the function capable of rendering many real-world processes logically by simplifying situations to non-negative results.
• This particular trait can be very useful in applied math, such as ensuring non-negative quantities like distance and speed.

Function Simplification

Function simplification is all about reducing expressions to their most basic form, making complex problems easier to manage. This is a fundamental skill used widely in solving mathematical problems because it streamlines the operations you must perform.
In our exercise, simplification helped to break down a composed function \((h \circ(g \circ f))(x)\) to one that's easier to handle and understand.
Starting with \(g(f(x))\), simplifying it from \((1/2)*(4x)+7\) to \(2x + 7\) removes any extraneous operations, allowing you to see the core of the function at work. Moreover, further simplifying \(h(g(f(x)))=|-4x -10|\) into a detailed piecewise format helps facilitate better insight on where and how the function operates.

• Breaking down functions can reveal hidden relationships and patterns, leading to deeper comprehension.
• It lays the foundation for more advanced topics that require critical thinking.

Function simplification isn't only about 'getting to the result'; it's about gaining mastery over the mathematics you are handling by reducing complexity wherever possible.