Problem 43 - College Algebra

Question

Express the given function h as a composition of two functions f and g so that \(h(x)=(f \circ g)(x)\) $$h(x)=|2 x-5|$$

Step-by-Step Solution

Answer

The function \(h(x)= |2x - 5|\) can be written as a composition of two functions \(f\) and \(g\) where \(f(x) = |x|\) and \(g(x) = 2x - 5\). So, \(h(x) = (f \circ g)(x)\).

Step 1: Define Function \(g\)

Consider \(g(x) = 2x - 5\). This represents the operation within the absolute value in the function \(h\).

Step 2: Define Function \(f\)

Consider \(f(x) = |x|\) since we simply take the absolute value of \(g(x)\) in the function \(h\).

Step 3: Verify the Composition

To ensure our choice of functions is correct, we'll check if \(f(g(x))\) gives us \(h(x)\). Replacing \(g(x)\) in \(f\), we get \(f(g(x)) = f(2x - 5) = |2x - 5|\). Thus, our choice of \(f\) and \(g\) were accurate because \(f(g(x)) = h(x)\).

Key Concepts

Absolute Value Function

The absolute value function plays an integral role in understanding how different mathematical expressions can be translated into comprehensible numbers representing distance, regardless of direction. It's written as \( f(x) = |x| \), where \( |x| \) denotes the absolute value of \( x \). This means it takes any number \( x \), and if \( x \) is negative, it converts \( x \) to positive, because, in terms of distance, direction is irrelevant.

For example, \( |3| \) equals 3 because it is already positive, while \( |-3| \) also equals 3, since the absolute value treats all numbers as if they were at the same distance from zero on a number line, but without the negative sign. This function's simplicity can be misleading because when we incorporate it into function composition, as shown in the exercise with \( h(x) = |2x - 5| \), it becomes evident that it can encapsulate more complex operations within itself.

Function Operations

Function operations include various ways in which functions can be combined to form new functions. These include addition, subtraction, multiplication, division, and composition of functions. Composition, symbolized by \( (f \text{circ} g)(x) \), is the process of applying one function to the results of another.

In simpler terms, when you have two functions, \( f(x) \) and \( g(x) \), composing them means applying \( g \) first to your input and then applying \( f \) to the result of \( g(x) \). This sequence is crucial: the output of \( g \) becomes the input for \( f \). This operation is central to our exercise, where we see the function \( h \) expressed as a composition of two functions, \( f \) and \( g \), where \( g \) modifies the input with a linear transformation and \( f \) subsequently applies the absolute value operation.

Verifying Function Composition

Verifying function composition is a vital step in understanding whether you've correctly combined functions to achieve a desired outcome. It involves substituting the output of one function into the next and checking if the final output matches the original complex function.

In the context of our exercise, after defining the functions \( f(x) = |x| \) and \( g(x) = 2x - 5 \), the verification comes by calculating \( f(g(x)) \) and observing if this matches the initial function \( h(x) = |2x - 5| \). If the result is in agreement, as is done in Step 3 of the solution, the composition is verified successfully. It essentially checks that the interior workings of the new function behave as expected when the inputs pass through the composed functions in the correct order.