02_Ex 1.2

Exercise Questions

Question	Link
Q1 the real valued function $f(x)$ and $g(x)$ are defined below…	1.2 Q-1
Q2. For the real valued function, $f$ defined below, find…	1.2 Q-2
Q3. Without finding the inverse, state the domain and range…	1.2 Q-3

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Overview

In this chapter, we explore the concept of function composition, which involves combining two functions to create a new function. Function composition allows us to apply one function to the result of another function. This chapter covers various compositions of functions, including $f \circ g(x)$, $g \circ f(x)$, and $f \circ f(x)$.

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Key Concepts

• Function Composition: The composition of two functions $f$ and $g$, denoted as $f \circ g(x)$, means applying $g(x)$ first and then applying $f$ to the result of $g(x)$. The formula is:
  $f \circ g(x) = f(g(x))$
• Inverse Composition: When $f$ and $g$ are inverses of each other, their compositions can simplify the expression.
• Domain Restrictions: The domain of the composition function is restricted by the domains of both $f$ and $g$. If any function has restrictions on its domain, the resulting composition will inherit those restrictions.

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Important Formulas

1. Composition of Functions:
   $f \circ g(x) = f(g(x))$
2. Example for $f(x) = 2x + 1$ and $g(x) = \frac{3}{x-1}$:
   $f \circ g(x) = f\left(\frac{3}{x-1}\right) = 2\left(\frac{3}{x-1}\right) + 1 = \frac{x+5}{x-1}$
3. Composition Example with Square Roots and Inverses:
   $f \circ g(x) = f\left(\frac{1}{x^2}\right) = \sqrt{x^2 + 1} / x$

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Tips and Tricks

• Simplifying Compositions: Always start by substituting the second function into the first function and simplifying step-by-step.
• Check for Domain Restrictions: Make sure to account for domain restrictions of both functions before simplifying the composition.
• Inverses: If $f$ and $g$ are inverses of each other, their composition will yield the identity function, i.e., $f(g(x)) = x$.

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Summary

This chapter focused on the composition of two functions, where we applied one function to the result of another. We explored different forms of function composition and worked through various examples. Key takeaways include the importance of understanding the domain restrictions and simplifying compositions step-by-step.

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Reference

By Sir Shahzad Sair:

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