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Chapter 2: Problem 43

Find \(f \circ g \circ h\) $$f(x)=x^{4}+1, \quad g(x)=x-5, \quad h(x)=\sqrt{x}$$

Short Answer

Expert verified
The composed function \(f \circ g \circ h(x) = (\sqrt{x} - 5)^4 + 1\).

Step by step solution

01

Understand the Composition

The composition of multiple functions, denoted as \(f \circ g \circ h\), means to apply one function to the result of another function's output. In this case, first apply \(h\), then \(g\), and finally \(f\).
02

Apply function h

Apply the function \(h(x) = \sqrt{x}\) to the variable \(x\). This means we take \(x\), and get \(h(x) = \sqrt{x}\).
03

Apply function g

Now that we have \(h(x) = \sqrt{x}\), substitute it into \(g(x) = x - 5\). This gives us \(g(h(x)) = g(\sqrt{x}) = \sqrt{x} - 5\).
04

Apply function f

Take the result from Step 3, \(g(h(x)) = \sqrt{x} - 5\), and substitute it into \(f(x) = x^4 + 1\). So, \(f(g(h(x))) = ((\sqrt{x} - 5)^4) + 1\).
05

Simplify the Expression

Calculate \((\sqrt{x} - 5)^4\). We expand this expression which involves binomial expansion, but the simplified form remains as \((\sqrt{x} - 5)^4\) depending on the level of simplification needed. The complete expression for \(f \circ g \circ h(x)\) is \((\sqrt{x} - 5)^4 + 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Composite Functions
Composite functions are like a series of actions that you perform one after the other. Think of it as baking a cake. First, you mix the batter, then bake it, and finally frost it. Here, when we find a composition such as \(f \circ g \circ h\), it tells us to apply function \(h\) first, then \(g\), and then \(f\). This order is crucial, just like following the right steps in a recipe.
  • Start with function \(h\).
  • Use its result as the input for function \(g\).
  • Finally, take the result of \(g\) and use it for \(f\).
Understanding compositions helps in tackling complex problems by breaking them into smaller, manageable tasks. It鈥檚 a powerful tool in mathematics for transforming expressions in a structured way.
Function Operations
Function operations are methods of using and combining functions, much like combining different ingredients in a dish. When you perform operations between functions, you're essentially substituting one mathematical rule for another.
  • Addition and subtraction: You can add or subtract entire functions, making new ones like \((f + g)(x) = f(x) + g(x)\).
  • Multiplication: Multiply functions to create combined effects, as in \((f \cdot g)(x) = f(x) \times g(x)\).
  • Composition: The focus here, where you use one function's output as another's input, as in \((f \circ g)(x) = f(g(x))\).
These operations allow for the flexible building of new functions from existing ones, enhancing the richness of mathematical expressions and solutions. By learning function operations, you get to manipulate functions to fit various problem-solving needs.
Mathematical Functions
Mathematical functions act like instructions or rules, mapping inputs to outputs in a consistent way. Each function takes an input (let's call it \(x\)) and gives an output according to a specific rule, like a vending machine giving a snack based on the button pressed.
  • Linear functions: Simple, straight-line rules like \(f(x) = x + 3\).
  • Quadratic functions: Include powers like \(f(x) = x^2 - x + 2\), forming parabolas when graphed.
  • Radical functions: Involve roots, like \(h(x) = \sqrt{x}\), which first appear in our composite sequence.
  • Complex functions: Combine these rules for more intricate results, as seen in the exercise with \(f(x) = x^4 + 1\).
Each type of function offers unique ways to process data and model real-world scenarios. Mastery of functions allows you to easily navigate through various mathematical landscapes, simplifying or solving problems with tailored approaches.

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Most popular questions from this chapter

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