Problem 8 - Applied Calculus

Question

Find the following: (a) \(f(g(x))\) (b) \(g(f(x))\) (c) \(f(f(x))\) $$f(x)=2 x^{2}\( and \)g(x)=x+3$$

Step-by-Step Solution

Answer

(a) \(2x^2 + 12x + 18\); (b) \(2x^2 + 3\); (c) \(8x^4\).

Step 1: Understanding the Composition of Functions

We are given two functions: \( f(x) = 2x^2 \) and \( g(x) = x + 3 \). The goal is to find the composition of these functions in different forms, specifically \( f(g(x)) \), \( g(f(x)) \), and \( f(f(x)) \). A function composition like \( f(g(x)) \) means we substitute the whole of \( g(x) \) into \( f(x) \).

Step 1: Evaluate \( f(g(x)) \)

Start by finding \( f(g(x)) \), which means replacing every \( x \) in \( f(x) \) with \( g(x) = x + 3 \). Thus, we have:\[ f(g(x)) = f(x + 3) = 2((x + 3)^2) \]Now, expand \((x + 3)^2\):\[(x + 3)^2 = x^2 + 6x + 9 \]Substitute back into the function, resulting in:\[ f(g(x)) = 2(x^2 + 6x + 9) = 2x^2 + 12x + 18 \]

Step 2: Evaluate \( g(f(x)) \)

Next, compute \( g(f(x)) \), which means substituting \( f(x) = 2x^2 \) into \( g(x) \):\[ g(f(x)) = g(2x^2) = 2x^2 + 3 \]Since \( g(x) = x + 3 \), we simply replace \( x \) by \( 2x^2 \).

Step 3: Evaluate \( f(f(x)) \)

Finally, find \( f(f(x)) \), which involves computing \( f \) applied to itself. This means substituting \( f(x) = 2x^2 \) into \( f(x) \): \[ f(f(x)) = f(2x^2) = 2((2x^2)^2) \]Simplifying \((2x^2)^2\) gives:\[(2x^2)^2 = 4x^4 \]Substitute back into \( f \), resulting in:\[ f(f(x)) = 2 imes 4x^4 = 8x^4 \]

Key Concepts

Composite Functions

Composite functions can be thought of as a series of operations performed in steps. You start with one function and then apply another function to the result of the first. In the exercise, you are working with two functions:

• \(f(x) = 2x^2\)
• \(g(x) = x + 3\)

The composition \(f(g(x))\) means you take the output of \(g(x)\) and use it as the input for \(f(x)\). The formula for function composition in general is \(f(g(x))\) or \(g(f(x))\).
You can think of it as a conveyor belt where each function represents a machine doing operations. Imagine \(g(x)\) as the first machine adding 3 to your input. Whatever comes out is then fed into the second machine, \(f(x)\), which processes it by squaring and then multiplying by 2.
This chaining of operations highlights the flexibility and power of function compositions, allowing for complex computations using simpler building blocks. Understanding function composition helps piece together how different functions interact and transform inputs through various stages.

Polynomial Expansion

Polynomial expansion involves transforming expressions like \((x+3)^2\) into a simpler form by multiplying out the terms. This is crucial when dealing with composite functions like \(f(g(x))\) because when you substitute one function into another, you often encounter expressions requiring expansion.
In our example, \(f(g(x)) = 2(x + 3)^2\). Expanding \((x+3)^2\) gives us \(x^2 + 6x + 9\). To do this, use the distributive property:

• Multiply \(x\) by \(x\), resulting in \(x^2\).
• Then \(2 \times x \times 3\), getting \(6x\).
• Finally, \(3 \times 3\) results in 9.

This gives us the expanded form: \(x^2 + 6x + 9\), which is then multiplied by 2 in the function \(f(x)\), resulting in the final expression \(2x^2 + 12x + 18\). Polynomial expansion simplifies the function evaluation process, making it easier to identify each term and factor in subsequent operations.

Quadratic Functions

Quadratic functions are a key part of the exercise, as indicated by \(f(x) = 2x^2\). Quadratic functions have the standard form \(ax^2 + bx + c\) with an essential characteristic being their U-shaped graph called a parabola.
They can represent diverse real-world phenomena like projectile motion or quadratic growth in populations. In the context of composite functions, quadratics add another layer of complexity because they involve squaring terms, which complicates both expansion and simplification.

• One attribute of quadratics is that they can either open upwards or downwards, determined by the sign of \(a\) (positive for upwards, negative for downwards). In \(f(x) = 2x^2\), \(a = 2\), so it opens upwards.
• The vertex of the parabola is either the lowest or highest point. Quadratics have symmetry around their vertical line called the axis of symmetry.

Recognizing the behavior of quadratics is crucial in interpreting functions within compositions and predicting outcomes of expressions. In operations like \(f(f(x))\), where you're nesting functions, the complexity can increase rapidly if not broken down into manageable steps.