Problem 45 - Precalculus with Limits

Question

In Exercises 41-48, find (a) \(f \circ g\), and (b) \(g \circ f\). Find the domain of each function and each composite function. \(f(x) = |x|\), \(g(x) = x + 6\)

Step-by-Step Solution

Answer

\(f \circ g(x) = |x + 6|\), \(g \circ f(x) = |x| + 6\), and the domain of both functions is all real numbers

Step 1: Find \(f \circ g\)

First, we find the function \(f \circ g\). This means we substitute \(g(x)\) into function \(f\). So, \(f(g(x)) = f(x + 6) = |x + 6|\)

Step 2: Find \(g \circ f\)

Now, we find the function \(g \circ f\). This means we substitute \(f(x)\) into function \(g\). So, \(g(f(x)) = g(|x|) = |x| + 6

Step 3: Find the domain of \(f \circ g\) and \(g \circ f\)

The domain of \(f \circ g\) and \(g \circ f\) are all real numbers, because the absolute value function and the linear function both accept all real numbers

Key Concepts

Function Composition

Understanding function composition is like learning how to combine two different recipes to create a new dish. When we have two functions, say function 'f' and function 'g', composing them creates a new function where we apply 'g' and take the result into 'f'. This operation is written as \(f \circ g\), which is read as 'f composed with g'.

For instance, consider \(f(x) = |x|\) and \(g(x) = x + 6\); interpreting composition, we replace 'x' in 'f' with \(g(x)\) to get \(f(g(x)) = f(x + 6) = |x + 6|\). Similarly, for \(g \circ f\), we input 'f(x)' into 'g' to get \(g(f(x)) = g(|x|) = |x| + 6\). It's like making a layered cake—start with one layer (g), add the next (f), and see the new, more complex creation.

Domain of a Function

The domain of a function is analogous to the ingredients list in a recipe; it's the complete set of possible 'inputs' for the function. If you input anything not on the list into the recipe, it might not work out.

When working out the domain for a composite function like \(f \circ g\) or \(g \circ f\), it's essential to ensure that every element in the domain of the 'outside' function maps to a valid input of the 'inside' function. Luckily, for our functions \(f(x) = |x|\) and \(g(x) = x + 6\), both have domains that include all real numbers, \(\mathbb{R}\), meaning you can put any real number into these functions, and they will give you a valid output. Therefore, the domain for \(f \circ g\) and \(g \circ f\) also includes all real numbers.

Absolute Value Function

The absolute value function is a function that takes any number as an input and returns its distance from zero on the number line, without regard to direction. This is symbolized by the vertical bars, like \(f(x) = |x|\). It's kind of like measuring how far you are from home, ignoring whether you traveled east or west to get there.

A key feature of the absolute value function is that it always outputs non-negative values. If you plug in \(x = -5\) or \(x = 5\), the absolute value function \(f(x) = |x|\) simplifies to 5 in both cases. This function's domain is also all real numbers because you can measure the distance from zero for any number on the number line.