Problem 7 - Thomas Calculus

Question

In Exercises \(7-10,\) write a formula for \(f \circ g \circ h\) $$ f(x)=x+1, \quad g(x)=3 x, \quad h(x)=4-x $$

Step-by-Step Solution

Answer

The formula for \(f \circ g \circ h\) is \(13 - 3x\).

Step 1: Understand the Composition

The composition of functions \(f \circ g \circ h\) means we must apply \(h\) first, then \(g\), and finally \(f\). This order is crucial when working with composed functions.

Step 2: Apply Function h

Start with applying \(h(x) = 4 - x\). For any input \(x\), \(h\) transforms it to \(4 - x\). Consider the output of this as \(h(x) = 4 - x\).

Step 3: Apply Function g to h(x)

Now apply \(g(x) = 3x\) to the output from the previous step. Substitute \(x = 4 - x\) (the result from \(h(x)\)) into \(g(x)\). Thus, we have:\[g(h(x)) = g(4-x) = 3(4-x) = 12 - 3x\].

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

Finally, apply \(f(x) = x + 1\) to the result of the previous composition, \(g(h(x)) = 12 - 3x\). Substitute this result into \(f(x)\). So, we have:\[f(g(h(x))) = f(12 - 3x) = (12 - 3x) + 1 = 13 - 3x\].

Step 5: Write the Complete Formula

The composed function \(f \circ g \circ h\) is the result of applying these functions in sequence from the inside out:\[(f \circ g \circ h)(x) = 13 - 3x\].

Key Concepts

Step-by-step Solution

Tackling function composition involves a sequential process, ensuring each function is applied correctly. Let's go through the steps:

• First, identify the functions involved and their order. Here, we have three: \(f(x) = x + 1\), \(g(x) = 3x\), and \(h(x) = 4 - x\).
• Understand that the notation \(f \circ g \circ h\) means you start with \(h\), apply \(g\) next, and finish with \(f\).

So, step by step, you compute:\[h(x) = 4 - x\]\[g(h(x)) = 3(4 - x) = 12 - 3x\]\[f(g(h(x))) = (12 - 3x) + 1 = 13 - 3x\].
This organized approach helps ensure accuracy and clarity when solving for composite functions.

Applying Multiple Functions

When dealing with multiple functions in composition, it's crucial to apply them in the correct order to avoid errors. Think of it as a series of transformations:

• Start by transforming your initial value with \(h(x)\).
• Then take that result and apply \(g(x)\).
• Lastly, use that new output with \(f(x)\).

Each function acts like a brick in a wall, where the output from one is the input for the next. It transforms the original input step by step. This sequential method, transforming one stage at a time, ensures each function is considered appropriately.

Order of Operations in Functions

The order of operations is a fundamental principle in function composition. It prevents misunderstandings and mistakes:

• Always start with the innermost function, moving outward. Begin with \(h(x)\) in our example.
• Apply each function to the result from the previous one. This method keeps transformations orderly and logical.
• Remember: function composition is not commutative, meaning \(f \circ g eq g \circ f\).

This precise order mirrors PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) in algebra but focuses on functions. It’s like a nested Russian doll, where you work on the inner piece before moving to the next outer layer.