91Ó°ÊÓ

The absolute value function is a mathematical way to express how far a number is from zero regardless of its direction on the number line. Simply put, it measures "magnitude" and not "direction." For example, both -3 and 3 have an absolute value of 3. This characteristic is written as \(|x|\). Here are some key points about the absolute value:

• It turns both positive and negative numbers into non-negative numbers.
• The absolute value of zero is zero.
• It is always expressed as a positive value or zero.

For a given number \(x\), you can mathematically express the absolute value as:\[|x| = \begin{cases} x, & \text{if } x \ge 0 \ -x, & \text{if } x < 0 \end{cases}\]This means that if \(x\) is positive or zero, the absolute value is \(x\). If \(x\) is negative, the absolute value is the opposite of \(x\). This mechanism ensures you never end up with a negative magnitude, which can be very useful in real-world applications such as determining distances.

A linear function is a simple yet powerful mathematical concept. It represents a straight line when graphed and has the general form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.In the context of the given problem, \(g(x) = 14x - 8\), it follows the linear form with:

• \(m = 14\) - the rate at which \(g(x)\) changes with \(x\).
• \(b = -8\) - the point where the line crosses the y-axis.

The slope \(m\) tells us how steep the line is. For example, a bigger slope means a steeper line. The y-intercept \(b\) indicates where the line crosses the y-axis, directly showing the value of \(y\) when \(x=0\). Linear functions are foundational in mathematics; they're easy to understand and widely used from basic algebra to complex data modeling.

Function composition involves inserting the output of one function directly into another. This concept can seem tricky, but breaking it down step-by-step clarifies the process.For \((f \circ g)(x)\):

• Start by evaluating \(g(x)\). In this case, \(g(x) = 14x - 8\).
• Then, plug this result into \(f(x)\). So, you are evaluating \(f(14x - 8)\), which translates to \(|14x - 8|\) because \(f(x) = |x|\).

For \((g \circ f)(x)\):

• Evaluate \(f(x)\) which gives \(|x|\).
• Plug this into \(g(x)\) to get \(g(|x|)\), leading to \(14|x| - 8\).

Through this step-by-step solution, students can clearly see how function outputs feed into one another, reinforcing both understanding and skill in combining different functions.