Question

Find (a) \(f \circ g\) and (b) \(g \circ f\). \(f(x)=|x|, \quad g(x)=x+6\)

Short answer

(a) \(f \circ g(x) = |x + 6|\)\n\n(b) \(g \circ f(x) = |x| + 6\)

Step-by-step solution

Compute \(f \circ g\)

To find the composition \(f \circ g(x)\), we simply substitute \(g(x)\) into \(f(x)\). Hence, \(f \circ g(x) = f(g(x)) = |g(x)| = |x + 6|\)

Compute \(g \circ f\)

To find the composition \(g \circ f(x)\), we substitute \(f(x)\) into \(g(x)\). Hence, \(g \circ f(x) = g(f(x)) = f(x) + 6 = |x| + 6\)

Key concepts

Understanding the Absolute Value Function

The absolute value function, denoted as \(|x|\), is a fundamental concept in mathematics. It represents the distance of a number from zero on the number line, regardless of its direction. Therefore, the absolute value of a number is always non-negative.

For example, \(|3| = 3\) and \(|-3| = 3\). No matter if the input is positive or negative, the output is always positive. The absolute value function is well-known for transforming negative numbers into their positive counterparts.

In the exercise, \( f(x) = |x| \), signifies that for every value of \(x\), regardless of its sign, the function will return the non-negative magnitude of \(x\). Its simplicity makes it an essential tool in solving real-world problems where distances and magnitudes are analyzed.

Arithmetics in Functions

Arithmetics in functions refers to basic mathematical operations performed within functions. When dealing with function arithmetics, operations such as addition, subtraction, multiplication, and division are commonly used.

Consider the function \(g(x) = x + 6\) from the original exercise. This is a linear function showing an arithmetic operation where 6 is added to the variable \(x\). Such linear adjustments are typical in algebra, making the function shift along the y-axis without altering its slope.

By applying arithmetics in functions, we modify existing functions to explore new relationships and model various mathematical scenarios. It simplifies complex problems, allowing for easier manipulation and understanding of functions.

Understanding Function Operations and Composition

Function operations allow us to create new functions by combining existing ones through addition, subtraction, multiplication, or division. Another powerful technique is function composition, which involves substituting one function into another.

Function composition is denoted as \((f \circ g)(x)= f(g(x))\), meaning the function \(g(x)\) is applied first, and then its result is substituted into function \(f(x)\). In the original exercise, for \(f \circ g(x) = |x + 6|\), we first modify \(x\) with \(g(x)=x+6\) before applying the absolute value function \(f\).

Similarly, \((g \circ f)(x)= g(f(x))\) implies that \(f(x)\) is calculated first, followed by evaluating \(g\) at this result. Thus, for \(g \circ f(x) = |x| + 6\), the absolute value \(f(x)\) gets computed first, then 6 is added as per \(g(x)\).

Function composition is versatile, as it allows multiple functions to work together to simplify or solve complex problems, transforming inputs and processes into manageable solutions.