Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-20: Find (a) \((f \circ g)(x)\) (b) \((g \circ f)(x)\) (c) \(f(g(-2))\) (d) \(g(f(3))\) $$ f(x)=3 x^{2}+4, \quad g(x)=5 x $$

Step-by-Step Solution

Answer

(a) \(75x^2 + 4\), (b) \(15x^2 + 20\), (c) 304, (d) 155.

Step 1: Understand the Functions

First, identify the given functions: \( f(x) = 3x^2 + 4 \) and \( g(x) = 5x \). These functions will be used to find compositions and function values.

Step 2: Composition of Functions for (a)

To find \((f \circ g)(x)\), substitute \(g(x)\) into \(f(x)\). This means that \((f \circ g)(x) = f(g(x)) = f(5x)\). Substitute \(5x\) into \(f(x) = 3x^2 + 4\): \(f(5x) = 3(5x)^2 + 4 = 3(25x^2) + 4 = 75x^2 + 4\).

Step 3: Composition of Functions for (b)

To find \((g \circ f)(x)\), substitute \(f(x)\) into \(g(x)\). This means that \((g \circ f)(x) = g(f(x)) = g(3x^2 + 4)\). Substitute \(3x^2 + 4\) into \(g(x) = 5x\): \(g(3x^2 + 4) = 5(3x^2 + 4) = 15x^2 + 20\).

Step 4: Evaluate Function for (c)

To find \(f(g(-2))\), first find \(g(-2)\). Use \(g(x) = 5x\):\(g(-2) = 5(-2) = -10\).Now substitute \(-10\) into \(f(x) = 3x^2 + 4\):\(f(-10) = 3(-10)^2 + 4 = 3(100) + 4 = 300 + 4 = 304\).

Step 5: Evaluate Function for (d)

To find \(g(f(3))\), first find \(f(3)\). Use \(f(x) = 3x^2 + 4\):\(f(3) = 3(3)^2 + 4 = 3(9) + 4 = 27 + 4 = 31\).Now substitute \(31\) into \(g(x) = 5x\):\(g(31) = 5(31) = 155\).

Key Concepts

Composite Functions

Composite functions are formed by combining two functions. This process blends one function with another to create a new outcome. Imagine each function as a machine. You place the result from the first machine (function) into the second one to get a final output.
For example, if you have two functions, say \( f(x) \) and \( g(x) \), like in this problem, you can create composite functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \). This means you'll substitute one function into the other.

• To find \((f \circ g)(x)\), think of it as putting \(g(x)\) into \(f(x)\).
• To find \((g \circ f)(x)\), do the reverse and substitute \(f(x)\) into \(g(x)\).

These compositions help analyze scenarios where one process feeds directly into another.

Evaluate Functions

Evaluating functions involves substituting numbers into the function's expression to calculate specific values. This is a simple, step-by-step process where you replace the variable \(x\) with a given number.

• First, identify the value you need to substitute. For instance, if you are calculating \(f(g(-2))\), begin by finding \(g(-2)\). This means replacing \(x\) in \(g(x)\) with \(-2\).
• Then, take the result from the first evaluation and plug it into the next function. So in our example, once you've found \(g(-2)\), use this value in \(f(x)\).

This method is incredibly useful in situations involving nested calculations. Encourage yourself to simplify neatly along the way!

Algebraic Functions

Algebraic functions are expressions made up of algebraic terms, which include operations like addition, subtraction, multiplication, and powers or roots.
The given functions in this problem, \( f(x) = 3x^2 + 4 \) and \( g(x) = 5x \), are straightforward examples of algebraic functions. Each term is a product of a constant and a power of \(x\).

• The power function in \(f(x)\) highlights a quadratic property because of the term \(3x^2\).
• The function \(g(x)\) is linear since it follows a simple multiplicative relationship, represented as \(5x\).

Understanding these basic structures helps in unraveling more complex expressions that build on these foundational ideas.

Nested Functions

Nested functions occur when one function is placed inside another, creating a layered effect. This nesting is visible in composite functions and during evaluations where outputs of one function become inputs to the next.
A classic example presented here is when performing tasks like \(f(g(-2))\) or \(g(f(3))\). Here, \(g(-2)\) leads to a value that nests within \(f(x)\), and vice versa.

• Begin with the innermost function. This isn't always obvious, so carefully follow the order in which they're applied.
• Work your way to the outer layers, using results from inner calculations as inputs for the next layer.

This layered processing is crucial in many real-world applications, where complex systems are simplified into a sequence of simpler, nested operations.