Problem 58 - Precalculus Essentials

Question

Find. a. \((f \circ g)(x)\) b. \((g \circ f)(x)\) c. \((f \circ g)(2)\) d. \((g \circ f)(2)\) $$f(x)=x^{2}+1, g(x)=x^{2}-3$$

Step-by-Step Solution

Answer

\((f \circ g)(x) = (x^{2}-3)^{2}+1,\) \((g \circ f)(x) = (x^{2}+1)^{2}-3,\) \((f \circ g)(2) = 4,\) and \((g \circ f)(2) = 12.\)

Step 1: Finding \((f \circ g)(x)\) and \((g \circ f)(x)\)

For \((f \circ g)(x)\), replace \(g(x)\) in the equation for \(f(x)\). This gives \(f(g(x)) = (x^{2}-3)^{2}+1\). This is your solution for \((f \circ g)(x)\). For \((g \circ f)(x)\), replace \(f(x)\) in the equation for \(g(x)\). This gives \(g(f(x)) = (x^{2}+1)^{2}-3\). This is your solution for \((g \circ f)(x)\).

Step 2: Evaluating \((f \circ g)(2)\) and \((g \circ f)(2)\)

Use the equations established from the initial step. For \((f \circ g)(2)\), substitute \(2\) into the equation to receive \((f \circ g)(2) = (2^{2}-3)^{2}+1 = 4\). For \((g \circ f)(2)\), substitute \(2\) into the equation to get \((g \circ f)(2) = (2^{2}+1)^{2}-3 = 12\).

Key Concepts

Function Composition

Function composition is an essential mathematical operation where two functions are combined to create a new function. It is akin to having two machines, where the output of the first machine feeds directly as the input for the second machine.
In mathematical notation, we write the composition of two functions, say \( f \) and \( g \), as \( (f \circ g)(x) \). This reads as "\( f \) of \( g \) of \( x \)". To find this, you apply \( g \) to \( x \) first and then apply \( f \) to the result from \( g(x) \).
For example, with the functions \( f(x) = x^2 + 1 \) and \( g(x) = x^2 - 3 \), to form \( (f \circ g)(x) \), replace each \( x \) in \( f \) with \( g(x) \), yielding \( f(g(x)) = (x^2 - 3)^2 + 1 \).

• This procedure transforms \( x \) gradually through both functions.
• The order in which functions are composed matters. \( (f \circ g)(x) \) can result in a very different function than \( (g \circ f)(x) \).

This technique is heavily used in calculus, computer programming, and complex systems, where multiple processes or rules need to be applied in sequence.

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input. Essentially, you're substituting a given number into the function to see what output you get, much like plugging different keys into a music box to hear what tune it plays.
Suppose we want to evaluate \( f(x) \) at \( x = 2 \). Given \( f(x) = x^2 + 1 \), substituting \( 2 \) into the function gives \( f(2) = 2^2 + 1 = 5 \).

• Evaluating functions can often help check specific results or validate sequences of transformations or operations.
• It is a direct point-by-point inspection, crucial for finding precise values in math problems.

Hence, in composite functions, like \( (f \circ g)(x) \), once you've found the composed function, say \( h(x) \), evaluating \( h(2) \) follows the same straightforward substitution method to yield an output.

Substitution Method

The substitution method in mathematics involves replacing one part of an equation or expression with a more suitable or simpler form. This approach plays a key role in problems involving composite functions by allowing us to simplify and tackle functions in manageable steps.
To illustrate with the example in our exercise, \( (f \circ g)(x) \) is evaluated by substituting \( g(x) \) into \( f(x) \), specifically substituting \( x^2 - 3 \) into each \( x \) of \( f \). This gives you the pathway for the complete expression that you then operate upon.

• Substitution is crucial when dealing with nested operations or complex functions.
• Each replacement should be tracked carefully to avoid errors, often using brackets to maintain clarity in operations.

This approach not only simplifies the operation but also helps in focusing on solving one piece of the problem at a time, enhancing clarity and precision.