Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(f(s)=\sqrt{s^{2}-4}\) and \(g(w)=|1+w|\), find formulas for \((f \circ g)(x)\) and \((g \circ f)(x)\).

Step-by-Step Solution

Answer

\( (f \circ g)(x) = \sqrt{x^2 + 2x - 3} \) and \( (g \circ f)(x) = |1 + \sqrt{x^2 - 4}| \).

Step 1: Understanding the Composition of Functions

Given two functions, we need to find \( (f \circ g)(x) \) and \( (g \circ f)(x) \). The notation \( (f \circ g)(x) \) means to apply \( g(x) \) first and then apply \( f\) to the result of \( g(x)\). On the other hand, \( (g \circ f)(x) \) means to apply \( f(x) \) first and then apply \( g\) to the result of \( f(x)\).

Step 2: Finding \( (f \circ g)(x) \)

To find \( (f \circ g)(x) \), substitute \( g(x)\) into \( f\). We have \( g(w) = |1 + w|\). So, \( g(x) = |1 + x|\). Now substitute into \( f(s)\): \\[ f(g(x)) = \sqrt{(|1+x|)^2 - 4} \] \This simplifies to \\[ \sqrt{1 + 2x + x^2 - 4} = \sqrt{x^2 + 2x - 3} \]

Step 3: Finding \( (g \circ f)(x) \)

To find \( (g \circ f)(x) \), substitute \( f(x)\) into \( g\). We have \( f(s) = \sqrt{s^2-4}\). So, substitute as follows: \\[ f(x) = \sqrt{x^2-4} \\] Now find \( g(f(x)) = |1 + \sqrt{x^2 - 4}|\). This is already simplified based on its definition.

Key Concepts

Mathematical Notation

Mathematical notation is a symbolic language that allows us to represent mathematical concepts in a structured and precise way. When working with complex functions, notation becomes incredibly important as it guides us through the process step by step. For example, in this exercise:

• \((f \circ g)(x)\) signifies the composition of functions \(f\) and \(g\). It tells us to substitute the output of \(g(x)\) as the input for \(f\).
• Similarly, \((g \circ f)(x)\) means we first evaluate \(f(x)\) and then use that as the input for \(g\).

The notation hints at the operational sequence and structure. By understanding this symbolic language, we can maintain clarity and precision as we transition from one function to another, simplifying the process of finding solutions.

Function Simplification

Function simplification involves reducing a complex expression into a more manageable and simpler form. This is crucial when analyzing composed functions, as it helps us understand their behavior more clearly.In the exercise, when we substitute \(g(x)\) into \(f\), the expression is:

• \(f(g(x)) = \sqrt{(|1+x|)^2 - 4}\)

By recognizing the squared term within the square root, we can remove the \(|1+x|\) to reveal the simplified form \(\sqrt{x^2 + 2x - 3}\). This simplification allows us to better understand the structure of the function without unnecessary complexity. Simplifying expressions in function composition not only makes them easier to evaluate but also unveils core properties that might not be evident at first sight.

Square Roots

Square roots are fundamental in mathematics, often used to "undo" squaring. They are symbolized by the radical sign, \(\sqrt{}\), and can be seen in various types of equations, particularly in quadratics and certain functions.In our exercise, the function \(f(s)=\sqrt{s^2-4}\) employs a square root which limits the domain to values where \(s^2 \geq 4\). Upon substituting \(g(x)\) into \(f\), a secondary simplification occurs as we resolve to \(\sqrt{x^2 + 2x - 3}\). Here, square roots ensure that we only consider real number outputs, preserving meaningful and applicable results.Understanding how to manipulate and simplify square roots is essential when examining the range and domain characteristics of functions. It aids in articulating the boundaries and behavior of function outputs.

Absolute Values

Absolute values measure the distance of a number from zero on a number line without considering which direction you're moving. It is denoted by two vertical bars \(|x|\). Absolute values are pivotal when composing functions as they determine how inputs are transformed without regard to sign.In our task, \(g(w)=|1+w|\) alters \(w\) to ensure only non-negative distances are considered. This property remains crucial when \(g(x)\) is placed into \(f\), as seen when we first evaluate \(|1+x|\). It directly affects the process of simplification by avoiding negative values, which could lead to incorrect or undefined results in some mathematical contexts.Overall, the absolute value ensures that only the magnitude affects subsequent computations, a key step when dealing with multi-layered composites like those in this exercise.