Problem 48 - Calculus Volume 1

Question

For each pair of functions, find a. \((f \circ g)(x)\) and b. \((g \circ f)(x)\) Simplify the results. Find the domain of each of the results. \(f(x)=|x+1|, g(x)=x^{2}+x-4\)

Step-by-Step Solution

Answer

(f ∘ g)(x) = |x² + x - 3| with domain (-∞, ∞); (g ∘ f)(x) = (|x+1|)² + |x+1| - 4 with domain (-∞, ∞).

Step 1: Find (f ∘ g)(x)

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). This means \(f(g(x)) = f(x^2 + x - 4)\). Since \(f(x) = |x+1|\), we have \[f(g(x)) = |(x^2 + x - 4) + 1| = |x^2 + x - 3|.\]

Step 2: Simplify (f ∘ g)(x)

The expression \(|x^2 + x - 3|\) is already in its simplest form because it involves an absolute value that cannot be reduced further.

Step 3: Determine the domain of (f ∘ g)(x)

Since \(f(x)\) is \(|x+1|\) and \(g(x)\) is a polynomial \(x^2 + x - 4\), and their composition \(f(g(x)) = |x^2 + x - 3|\) involves an absolute value of a polynomial, it is defined for all real numbers. Therefore, the domain is \((-\infty, \infty).\)

Step 4: Find (g ∘ f)(x)

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). This means \(g(f(x)) = g(|x+1|)\). Since \(g(x) = x^2 + x - 4\), we have \[g(f(x)) = (|x+1|)^2 + |x+1| - 4.\]

Step 5: Simplify (g ∘ f)(x)

The expression \((|x+1|)^2 + |x+1| - 4\) cannot be simplified further as it involves absolute values and polynomial terms.

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

The function \(g(f(x)) = (|x+1|)^2 + |x+1| - 4\) is defined for all \(x\) because it involves polynomials and absolute values, which are defined for all real numbers. Therefore, the domain is \((-\infty, \infty).\)

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when working with function composition. The domain represents all the values that can be input into the function to produce a valid output. In simpler terms, it's the set of all possible 'x' values that you can plug into the function without causing any mathematical issues like division by zero or taking the square root of a negative number.

For polynomial functions, such as \(g(x) = x^2 + x - 4\), the domain is all real numbers \((-\infty, \infty)\) because polynomials are defined everywhere on the real number line. Similarly, absolute value functions like \(f(x) = |x+1|\) are also defined for all real numbers.

When you compose these functions— either \((f \circ g)(x)\) or \((g \circ f)(x)\)— the resulting functions remain defined for all real numbers. This is why, in both cases of our exercise, the domain is \((-\infty, \infty)\). Whether you start with a polynomial or an absolute value function, the composition does not impose any new restrictions on the domain.

Absolute Value Function

An absolute value function is a special kind of function that outputs the non-negative value of a number. It's represented as \(|x|\), which equals \(x\) if \(x\) is non-negative, and \(-x\) if \(x\) is negative. Absolute value functions are useful in many mathematical contexts because they measure distance without regard to direction.

In the function \(f(x) = |x+1|\), it takes any input \(x\), adds 1 to it, and then outputs the distance of this result from zero. Absolute value functions are significant in composition because they handle their inputs differently compared to regular functions. For example, in the composition \((f \circ g)(x) = |x^2 + x - 3|\), the polynomial \(g(x)\) is modified to ensure a non-negative output.

The presence of an absolute value means that the structure stays versatile enough to handle any input value, which simplifies finding the domain. Every real number will be accepted as an input, emphasizing the function's comprehensive nature.

Polynomial Functions

Polynomial functions, like \(g(x) = x^2 + x - 4\), play a fundamental role in many areas of mathematics due to their straightforward properties and easy manipulation. They are sums of terms consisting of variables raised to non-negative integer powers multiplied by coefficients, and are defined for all real numbers.

These functions are predictable: you can always plug in any real number and expect a real output because there are no operations that can create undefined conditions, such as division by zero or taking the square root of a negative number. In our exercise, when we evaluate \(g(x)\) or compose it with \(f(x)\) to form \((g \circ f)(x) = (|x+1|)^2 + |x+1| - 4\), the polynomial form ensures that the function remains well-defined over its entire domain.

The simplicity and broad domain of polynomial functions make them an excellent choice for function composition. When combined with other functions, like absolute value functions, they add flexibility without narrowing the scope of input values.