Problem 36 - Algebra and Trigonometry

Question

\(29-40\) Find the functions \(f \circ g, g \circ f, f \circ f,\) and \(g \circ g\) and their domains. $$ f(x)=x-4, \quad g(x)=|x+4| $$

Step-by-Step Solution

Answer

All compositions have the domain \((-\infty, \infty)\): \(f(g(x)) = |x+4| - 4\), \(g(f(x)) = |x|\), \(f(f(x)) = x - 8\), \(g(g(x)) = ||x+4| + 4|\).

Step 1: Compute f(g(x))

To find the function composition \(f(g(x))\), substitute \(g(x)\) into \(f(x)\). Start by replacing \(x\) in \(f(x) = x - 4\) with \(g(x) = |x + 4|\). This results in: \[ f(g(x)) = |x+4| - 4 \]

Step 2: Determine the domain of f(g(x))

The domain of \(f(g(x))\) is determined by the domain of \(g(x) = |x+4|\), which is all real numbers. Since the function \(f(x) = x - 4\) is also defined for all real numbers, the domain of \(f(g(x))\) is all real numbers \( (-\infty, \infty) \).

Step 3: Compute g(f(x))

To compute \(g(f(x))\), substitute \(f(x)\) into \(g(x)\). Replace \(x\) in \(g(x) = |x + 4|\) with \(f(x) = x - 4\):\[ g(f(x)) = |(x - 4) + 4| = |x| \]

Step 4: Determine the domain of g(f(x))

The function \(g(f(x)) = |x|\) is defined for all real numbers because the absolute value function \(|x|\) can take any real number as input. Therefore, the domain of \(g(f(x))\) is \((-\infty, \infty)\).

Step 5: Compute f(f(x))

For the function composition \(f(f(x))\), substitute \(f(x)\) into itself. Replace \(x\) in \(f(x) = x - 4\) with \(f(x) = x - 4\):\[ f(f(x)) = (x - 4) - 4 = x - 8 \]

Step 6: Determine the domain of f(f(x))

The function \(f(f(x)) = x - 8\) is a linear function, which is defined for all real numbers. Hence, the domain of \(f(f(x))\) is \((-\infty, \infty)\).

Step 7: Compute g(g(x))

To find \(g(g(x))\), substitute \(g(x)\) into itself. Replace \(x\) in \(g(x) = |x + 4|\) with \(g(x) = |x + 4|\):\[ g(g(x)) = ||x + 4| + 4| \]

Step 8: Determine the domain of g(g(x))

\(g(g(x)) = ||x + 4| + 4|\) involves nested absolute values. Since absolute values are defined for all real numbers, the domain of \(g(g(x))\) is \((-\infty, \infty)\).

Key Concepts

Domain of a Function

Understanding the domain of a function is essential in math. It tells us all the possible input values (or x-values) for which the function is defined. Imagine it as the set of all "starting points" you can use in a function without breaking its rules.
For example, when dealing with polynomial functions like linear functions, the domain is typically all real numbers, denoted as (-\infty, \infty)\. These functions can accept any real number.

• For the linear function \(f(x) = x - 4\), the domain is all real numbers.

Absolute value functions, such as \(g(x) = |x+4|\), also have a domain of all real numbers. This is because the absolute value operation is defined for every real number. It essentially measures how far a number is from zero without any sign. The domain of a composition like \(f(g(x))\) or \(g(f(x))\) can be examined by combining the individual domains of the functions involved. If both functions accept all real numbers, the composition wil also have a domain of all real numbers, hence (-\infty, \infty)\.

• These include the compositions \(f(g(x))\), \(g(f(x))\), and \(g(g(x))\).

Absolute Value Function

The absolute value of a number is always a non-negative number. It tells us how far a number is from zero on the number line, ignoring its sign. This is denoted by vertical bars around the number or expression, for example, \(|x|\).

• Absolute values help create functions like \(g(x) = |x + 4|\).

When working with absolute value functions, they can sometimes wrap other expressions inside the vertical bars, ensuring the result is always non-negative. They can also layer over themselves, as seen in a function like \(g(g(x)) = ||x + 4| + 4|\), where one absolute value operation is applied over another.
It's important to remember:

• The absolute value function is always defined for all real numbers.
• It has a range starting from zero and extending to positive infinity.

Linear Function

Linear functions are one of the simplest and most fundamental types of functions. They form straight lines on a graph. A linear function has the general form \(f(x) = ax + b\), where \(a\) and \(b\) are constants, and \(x\) is the variable. This form means it changes at a constant rate.

• In this exercise, the function \(f(x) = x - 4\) is a linear function where \(a = 1\) and \(b = -4\).

Because linear functions continue indefinitely in both directions along the x-axis, their domain is always all real numbers, (-\infty, \infty)\. These functions are very versatile and appear frequently in different branches of mathematics.
When composing linear functions with other types of functions, like in \(f(f(x)) = x - 8\), the characteristic form remains, changing only by the constant term. This maintains the domain as (-\infty, \infty)\.