Problem 46 - 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. \(f(x) = |x-4|\), \(g(x) = 3-x\)

Step-by-Step Solution

Answer

The composite function \(f \circ g(x) = |-x - 1|\) and its domain is all real numbers. The composite function \(g \circ f(x) = 3 - |x - 4|\) and its domain is also all real numbers. The domain of \(f(x) = |x -4|\) and \(g(x) = 3-x\) is all real numbers.

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

Substitute \(g(x)\) into \(f(x)\) for \(x\). Thus, \(f \circ g(x) = f(g(x)) = |g(x) - 4| = |3 - x - 4| = |-x -1|.\n

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

Substitute \(f(x)\) into \(g(x)\) for \(x\). So, \(g \circ f(x) = g(f(x)) = 3 - f(x) = 3 - |x - 4|\.

Step 3: Find the domain of each function

The domain of a function is the set of input for which the function is defined. The domain of \(f(x) = |x - 4|\) is all real numbers because any number subtracted by 4 remains a real number and the absolute value also returns a real number. Thus, the domain of \(f(x)\) is \(-\infty < x < \infty\) or \(\mathbb{R}\). For the function \(g(x) = 3 - x\), any real number substracted from 3 also provides a real number. Hence, the domain of \(g(x)\) is also \(-\infty < x < \infty\) or \(\mathbb{R}\). The domain of the composite function \(f \circ g(x) = |-x -1|\) is also all real numbers because any real number inverted and subtracted by 1 will also be a real number and the absolute value of it will also be a real number. Finally, for \(g \circ f(x) = 3 - |x - 4|\), it too has the domain of all real numbers for the same reasons.\n

Key Concepts

Function Composition

Understanding how functions combine is essential, particularly in the context of function composition. In simpler terms, function composition is the process of applying one function to the results of another. For example, if you have two functions, say, f(x) and g(x), the composite function f \( \circ \) g means you first apply g(x), then take that result and apply f(x) to it.

The process can be visualized like a two-step journey: starting at x, moving to g(x), and finally landing at f(g(x)). It's important to take note of the order, as f \( \circ \) g may yield a different result than g \( \circ \) f. In the given exercise, we saw how to compute each composition and noted that the order of functions matters because |-x -1| is not the same as 3 - |x - 4|.

In practice, be sure to substitute the entire expression from the inner function where the variable x appears in the outer function. This method aids in maintaining accuracy when dealing with more complex expressions or functions.

Domain of a Function

The domain of a function represents all possible values that can be input into the function and is foundational when examining how functions behave. In technical terms, it's the set of all 'x-values' that will output real 'y-values'. Whenever you determine the domain, consider the limitations: fractions can't have zero denominators, square roots can't have negative radicands, and logarithms can't have non-positive arguments.

With respect to absolute value functions, like the one in our example f(x) = |x - 4|, the domain includes all real numbers because no matter what value 'x' is, the absolute value ensures a real number result. The same applies to linear functions, like g(x) = 3 - x, which inherently accept all real numbers. As a crucial step, we always confirm that the composite functions derived also have domains consisting of all real numbers, as seen in both f \( \circ \) g(x) and g \( \circ \) f(x), which handle real numbers without restriction.

Absolute Value

The concept of absolute value is profoundly simple yet powerful. It's the distance from zero on the number line, and it's always non-negative. Mathematical notation uses two vertical bars, like this: |x|. The absolute value of a number or an expression tells us how far away it is from zero without bothering about the direction.

This attribute makes the absolute value incredibly useful in various areas of mathematics, including function composition. In our exercise, we dealt with the absolute value in f(x) = |x - 4| and its composition with g(x). The key takeaway is that no matter what the input is, whether it's a direct value of x or an expression resulting from another function, taking the absolute value ensures that the output will be non-negative. Understanding absolute value can significantly help in grasping how functions behave, especially when they interact through composition.