Problem 9 - Thomas Calculus

Question

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=\sqrt{x+1}, \quad g(x)=\frac{1}{x+4}, \quad h(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The formula for \( f \\circ g \\circ h(x) \) is \( \sqrt{\frac{5x + 1}{1 + 4x}} \).

Step 1: Understand Function Composition

Function composition is the process of applying one function to the result of another function. Here, we need to find the composition of three functions: \[ (f \circ g \circ h)(x) = f(g(h(x))) \] This means we will first apply the function \( h(x) \), then take that result and apply \( g(x) \), and finally apply \( f(x) \) to the result of the previous composition.

Step 2: Apply Function h(x)

First, we apply the function \( h(x) = \frac{1}{x} \):\[ h(x) = \frac{1}{x} \] This is the innermost function in the composition.

Step 3: Apply Function g(x) on h(x)

Next, we substitute \( h(x) \) into \( g(x) \):\[ g(h(x)) = g\left(\frac{1}{x}\right) = \frac{1}{\left(\frac{1}{x}\right) + 4} \]Simplifying the expression gives:\[ = \frac{x}{1 + 4x} \] This represents \( g(h(x)) \).

Step 4: Apply Function f(x) on g(h(x))

Finally, substitute \( g(h(x)) \) into \( f(x) \):\[ f(g(h(x))) = f\left(\frac{x}{1 + 4x}\right) = \sqrt{\frac{x}{1 + 4x} + 1} \]To simplify:\[ = \sqrt{\frac{x + (1 + 4x)}{1 + 4x}} \]Simplifying inside the square root:\[ = \sqrt{\frac{5x + 1}{1 + 4x}} \]This is the formula for \( f \circ g \circ h(x) \).

Key Concepts

Composition of Functions

The composition of functions is a fascinating concept that allows us to create more complex functions by combining simpler ones. Imagine you have three different mathematical functions: a square root function, a fraction function, and another fraction function. By composing these, you can form a single function that captures all their behaviors.
In function composition, you apply one function to the result of another, progressively building up a new function. For example, with functions \( f(x) \), \( g(x) \), and \( h(x) \), you can create \((f \circ g \circ h)(x)\), which is equivalent to \(f(g(h(x)))\).
This process is like solving a puzzle where each piece is a function, and together they form a new picture of mathematical relationships.
When starting with \( h(x) \), you take that output and apply \( g(x) \), and finally, you use the result to apply \( f(x) \). This sequence of actions makes the composition of functions both powerful and versatile in analyzing function behaviors.

Mathematical Functions

Mathematical functions are like machines in mathematics. They take an input, do some processing based on their rule, and give an output. Each function has its own specific rule or formula.
Let's break down the functions used in this exercise:

• \( f(x) = \sqrt{x+1} \): This function adds 1 to the input \( x \) and then takes the square root of the result.
• \( g(x) = \frac{1}{x+4} \): This function shifts the input \( x \) by adding 4 and then takes the reciprocal of the result.
• \( h(x) = \frac{1}{x} \): This function simply takes the reciprocal of the input \( x \).

Each of these functions manipulates the input differently, and in function composition, understanding each process is vital. It is essential to apply them in the correct order to get the desired result. Knowing how each machine works separately makes it easier to see how their combination will work.

Step-by-Step Solution

To solve a problem using function composition, we follow a systematic step-by-step approach. Let's go through the procedure used to tackle this specific exercise:
### Step 1: Apply Function \( h(x) \)
Start with the innermost function. Here, we have \( h(x) = \frac{1}{x} \). This will be our initial input.
### Step 2: Apply Function \( g(x) \) on \( h(x) \)
Next, take the result from \( h(x) \) and apply it to \( g(x) \). Substitute \( h(x) \) into \( g(x) \):
\[ g\left(h(x)\right) = g\left(\frac{1}{x}\right) = \frac{x}{1 + 4x} \]
### Step 3: Apply Function \( f(x) \) on \( g(h(x)) \)
Finally, we use the result from \( g(h(x)) \) and apply it in \( f(x) \):
\[ f(g(h(x))) = f\left(\frac{x}{1 + 4x}\right) = \sqrt{\frac{x + (1 + 4x)}{1 + 4x}} \]
After simplifying, we arrive at:
\[ = \sqrt{\frac{5x + 1}{1 + 4x}} \]
This step-by-step breakdown is crucial for understanding how different functions work together to form a new, composite function. Following each step methodically ensures clarity and helps avoid mistakes.