Problem 7 - Thomas Calculus

Question

If \(u(x)=4 x-5, v(x)=x^{2},\) and \(f(x)=1 / x,\) find formulas for the following. $$ \begin{array}{ll}{\text { a. } u(v(f(x)))} & {\text { b. } u(f(v(x)))} \\\ {\text { c. } v(u(f(x)))} & {\text { d. } v(f(u(x)))} \\ {\text { e. } f(u(v(x)))} & {\text { f. } f(v(u(x)))}\end{array} $$

Step-by-Step Solution

Answer

a) \(u(v(f(x))) = \frac{4}{x^2} - 5\); b) \(u(f(v(x))) = \frac{4}{x^2} - 5\); c) \(v(u(f(x))) = (\frac{4}{x} - 5)^2\); d) \(v(f(u(x))) = (\frac{1}{4x-5})^2\); e) \(f(u(v(x))) = \frac{1}{4x^2 - 5}\); f) \(f(v(u(x))) = \frac{1}{(4x-5)^2}\).

Step 1: Simplify u(v(f(x)))

To find \(u(v(f(x)))\), start by finding \(f(x) = \frac{1}{x}\).\[v(f(x)) = v\left(\frac{1}{x}\right) = \left(\frac{1}{x}\right)^2\]Then, substitute \(v(f(x))\) into \(u(x)\):\[u(v(f(x))) = u\left(\frac{1}{x^2}\right) = 4\left(\frac{1}{x^2}\right) - 5 = \frac{4}{x^2} - 5\]

Step 2: Simplify u(f(v(x)))

To find \(u(f(v(x)))\), follow these steps:\[v(x) = x^2\]\[f(v(x)) = f(x^2) = \frac{1}{x^2}\]Input into \(u(x)\):\[u(f(v(x))) = u\left(\frac{1}{x^2}\right) = 4\left(\frac{1}{x^2}\right) - 5 = \frac{4}{x^2} - 5\]

Step 3: Simplify v(u(f(x)))

For \(v(u(f(x)))\):\[f(x) = \frac{1}{x}\]\[u(f(x)) = u\left(\frac{1}{x}\right) = 4\left(\frac{1}{x}\right) - 5 = \frac{4}{x} - 5\]With this, find:\[v(u(f(x))) = v\left(\frac{4}{x} - 5\right) = \left(\frac{4}{x} - 5\right)^2\]

Step 4: Simplify v(f(u(x)))

For \(v(f(u(x)))\):\[u(x) = 4x - 5\]\[f(u(x)) = f(4x - 5) = \frac{1}{4x - 5}\]Find:\[v(f(u(x))) = v\left(\frac{1}{4x - 5}\right) = \left(\frac{1}{4x - 5}\right)^2\]

Step 5: Simplify f(u(v(x)))

For \(f(u(v(x)))\):\[v(x) = x^2\]\[u(v(x)) = u(x^2) = 4x^2 - 5\]Finally:\[f(u(v(x))) = f(4x^2 - 5) = \frac{1}{4x^2 - 5}\]

Step 6: Simplify f(v(u(x)))

For \(f(v(u(x)))\):\[u(x) = 4x - 5\]\[v(u(x)) = v(4x - 5) = (4x - 5)^2\]Resulting in:\[f(v(u(x))) = f((4x - 5)^2) = \frac{1}{(4x - 5)^2}\]

Key Concepts

Algebraic Functions

Algebraic functions play a crucial role in mathematics, as they involve operations like addition, subtraction, multiplication, division, and raising to power, applied to variable expressions. In the exercise provided, we explore three different algebraic functions:

• The linear function, \(u(x) = 4x - 5\), which offers a simple transformation representing a line on the Cartesian plane.
• The quadratic function, \(v(x) = x^2\), characterized by its squared term, forming a parabola.
• The reciprocal function, \(f(x) = \frac{1}{x}\), which inverts the input and results in a hyperbola.

Each of these functions involves different operations which can simplify or expand expressions.
Recognizing the form of these functions helps in manipulating and combining them.Thus, understanding their individual properties is essential for tackling problems that involve composition, like the exercise we are solving.

Composite Functions

Composite functions are created when the output of one function becomes the input of another. This concept is essential to understand how complex functions are structured from simpler ones. In the given exercise, we observe various combinations where functions are layered upon one another:

• The notation \(u(v(f(x)))\) signifies the application of \(f(x)\), followed by \(v(x)\), and finally \(u(x)\).
• A composite function, such as this, necessitates substituting the result of one function into the variable position of the next, continually solving until reaching a final simplified expression.

To master composite functions, it's vital to work methodically through each function step-by-step.
This not only aids in ensuring accuracy but also enhances understanding of how inner transformations affect those performed later.Composite functions expand the toolkit of mathematical operations, allowing expressions to be manipulated in more dynamic ways.

Step-by-Step Solution

Solving complex mathematical expressions often requires breaking down the process into manageable steps. This step-by-step approach is especially useful when dealing with composite functions, as seen in the exercise.

• Start by addressing the innermost function and solve it first, then use this result as input for the next function.
• For example, in \(u(v(f(x)))\): calculate \(f(x) = \frac{1}{x}\), then \(v\left(f(x)\right) = \left(\frac{1}{x}\right)^2\), and finally apply \(u\left(v(f(x))\right)\) to get \(u\left(\frac{1}{x^2}\right) = \frac{4}{x^2} - 5\).
• This clear, sequential approach ensures each transformation is correct before proceeding, simplifying the complexity inherent in function composition.

By visualizing and logically ordering the steps, students can enhance their problem-solving skills and gain confidence in handling algebraic tasks.
This methodical progression is invaluable when tackling more intricate mathematical problems.