91Ó°ÊÓ

Understanding the absolute value function is crucial for solving a range of mathematical problems. Essentially, the absolute value of a number represents its distance from zero on the number line, regardless of direction. Mathematically, for any real number x, the absolute value is defined as:
\[|x| = \begin{cases} x & \text{if } x\geq 0, \ -x & \text{if } x<0. \end{cases}\]
This piecewise definition shows that if x is positive or zero, the absolute value of x is just x itself. If x is negative, however, the absolute value of x is the opposite of x.

When it comes to functions, the absolute value function takes any input and delivers a non-negative output. In our problem, the function \( h(x) = |3x - 4| \) involves an absolute value, indicating that for any real number x, the output is the distance of \( 3x - 4 \) from zero. This function will flatten out any negative results from the equation \( 3x - 4 \), making the graph of this function a 'V' shape centered at the point where \( 3x - 4 = 0 \).

A linear function is another fundamental concept in algebra and represents a straight line when graphed on a coordinate plane. The general form of a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope \( m \) dictates the steepness and direction of the line, while the y-intercept \( b \) indicates where the line crosses the y-axis.

In the textbook problem, the inner function \( g(x) = 3x - 4 \) is a linear function. Here, the slope is 3, and the y-intercept is -4. This linear function will increase by 3 units in the y direction for every 1 unit increase in the x direction. When we include this linear function within an absolute value function, it is transformed, with the parts of the line that would normally be below the x-axis reflected upwards to create the 'V' shape mentioned previously.

Function decomposition involves breaking down a complex function into simpler, constituent functions. It can be particularly handy when dealing with composite functions. A composite function \( h(x) = (f \circ g)(x) \) is formed when one function \( f \) is applied to the result of another function \( g(x) \).

Returning to our example, the function \( h(x) = |3x - 4| \) can be decomposed into two simpler functions. We identified the linear function \( g(x) = 3x - 4 \) as the inner function. This is what we 'feed' into the outer function first. The absolute value function \( f(x) = |x| \) acts as the outer function, processing the output of \( g(x) \). By applying function composition, we verify the accuracy of our decomposition: \( f(g(x)) = |3x - 4| \), which matches the original composite function \( h(x) \). Understanding and applying function decomposition allows us to tackle complex mathematical problems more effectively by addressing their components one step at a time.