Problem 61 - Calculus: One and Several Variables

Question

Set \(f(x)=x^{2}-4, g(x)=\frac{3 x}{2-x}, h(x)=\) \(\sqrt{x + 4},\) and \(k(x)=\frac{2 x}{3+x} .\) Use a CAS to find the indicated composition. (a) \(f \circ g ;\) (b) \(g \circ k ;\) (c) \(f \circ k \circ g\)

Step-by-Step Solution

Answer

\[(a) f(g(x)) = \left(\frac{3x}{2-x}\right)^{2} - 4\]\[(b) g(k(x)) = \frac{3(\frac{2x}{3+x})}{2-\frac{2x}{3+x}}\]\[(c) f(k(g(x))) = \left(\frac{2(\frac{3x}{2-x})}{3+\frac{3x}{2-x}}\right)^{2} - 4\]

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

To find \(f \circ g\), replace every 'x' in function \(f(x)\) with the function \(g(x)\).\nSo, \(f(g(x)) = g(x)^{2} - 4 = \left(\frac{3x}{2-x}\right)^{2} - 4.\nThis is the composition of \(f\) and \(g\). Simplifying, you can leave it as is or calculate further based on whether we want a simplified form or not.

Step 2: Calculate \(g \circ k\) (g of k)

To find \(g \circ k\), the function \(g(x)\) has 'x' replaced with \(k(x)\).\nSo, \(g(k(x)) = \frac{3k(x)}{2-k(x)} = \frac{3(\frac{2x}{3+x})}{2-\frac{2x}{3+x}}.\nThis is the composition of \(g\) and \(k\). Again, leave the expression as is, or you may simplify based on what the actual task requires.

Step 3: Calculate \(f \circ k \circ g\) (f of k of g)

To find \(f \circ k \circ g\), replace each 'x' in \(f(x)\) with \(k(g(x))\).\nSo, \(f(k(g(x))) = k(g(x))^{2} - 4 = \left(\frac{2g(x)}{3+g(x)}\right)^{2} - 4 = \left(\frac{2(\frac{3x}{2-x})}{3+\frac{3x}{2-x}}\right)^{2} - 4\nThis is the composition of \(f\), \(k\) and \(g\). Again, you can leave it as is or simplify the function depending on the specifics of the exercise.

Key Concepts

Mathematical Functions

Mathematical functions are fundamental in mathematics, serving as the bridge between different quantities. Simply put, a function maps each input from its domain to an output in its range. This concept explains how variables depend on each other.

• Understanding Notation: In functions like \(f(x)=x^2-4\), the expression "\(x^2-4\)" describes how "\(x\)" is transformed into an output.
• Domain and Range: The domain is the set of all possible inputs, while the range is the set of all possible outputs. For instance, in \(g(x)=\frac{3x}{2-x}\), the domain excludes where the denominator is zero.
• Transforming Variables: Functions allow us to substitute variables to explore diverse mathematical scenarios, leading to the "composing" concept.

In essence, understanding functions involve grasping how inputs are systematically converted into outputs using specific rules.

Calculus Problems

Calculus is all about change and motion, and often requires working with equations and functions to determine how things vary. Problems involving calculus frequently utilize derivatives and integrals.

• Derivatives: Involves finding the rate at which a function is changing at any given point. This is crucial in problems that need slope or speed calculations.
• Integrals: Used for calculating areas under curves, which implies finding the total quantity where rate is known.

In the context of solving calculus problems, recognizing how functions interact and change can help predict outcomes and solve equations efficiently.

Composite Functions

When we talk about composite functions, we're dealing with scenarios where multiple functions are combined so that the output of one function becomes the input of another. This concept is commonly denoted as \(f \circ g\), read as "\(f\) of \(g\)".

• Formation: To create a composite function like \(f \circ g\), you evaluate \(g(x)\) first and then use its output as input for \(f(x)\).
• Simplification: Simplifying composite functions often involves algebraically manipulating expressions for a cleaner or more usable form.
• Application: Composite functions help solve complex problems by breaking them into simpler, manageable parts and recalculating outputs through sequential processes.

In essence, composites reveal the interplay between different functions, expanding their application and utility in mathematics.

Geometry of Functions

The geometry of functions refers to the graphic representation of functions on a coordinate plane, describing how functions appear visually.

• Graphing Functions: Each mathematical function can be plotted on a graph, where the x-axis represents the input, and the y-axis represents the output.
• Visual Analysis: By graphing, you can spot critical behavior like intercepts, max/min points, and asymptotic behavior visually.
• Understanding Compositions: Composite functions can also be graphed, depicting how chains of functions work together visually, aiding in comprehension of functions we interact with.

Graphing provides an intuitive way to explore and understand functions beyond symbolic and algebraic manipulations.