Question

For each pair of functions, \(f(x)\) and \(g(x)\), determine \(f(g(x))\) and \(g(f(x))\). a) \(f(x)=x^{2}+x\) and \(g(x)=x^{2}+x\) b) \(f(x)=\sqrt{x^{2}+2}\) and \(g(x)=x^{2}\) c) \(f(x)=|x|\) and \(g(x)=x^{2}\)

Short answer

For (a), both \( f(g(x)) \) and \( g(f(x)) \) are \( x^4 + 2x^3 + 2x^2 + x \). For (b), \( f(g(x)) = \sqrt{x^4 + 2} \) and \( g(f(x)) = x^2 + 2 \). For (c), both \( f(g(x)) \) and \( g(f(x)) \) are \( x^2 \).

Step-by-step solution

Determine f(g(x)) for a

Given: \( f(x) = x^2 + x \) \( g(x) = x^2 + x \)To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):\[ f(g(x)) = f(x^2 + x) \]Now replace every \( x \) in \( f(x) \) with \( x^2 + x \):\[ f(x^2 + x) = (x^2 + x)^2 + (x^2 + x) \]Expand and simplify:\[ (x^2 + x)^2 = x^4 + 2x^3 + x^2 \]Therefore:\[ f(g(x)) = x^4 + 2x^3 + x^2 + x^2 + x \]Combine like terms:\[ f(g(x)) = x^4 + 2x^3 + 2x^2 + x \]

Determine g(f(x)) for a

To find \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \):\[ g(f(x)) = g(x^2 + x) \]Now replace every \( x \) in \( g(x) \) with \( x^2 + x \):\[ g(x^2 + x) = (x^2 + x)^2 + (x^2 + x) \]This yields the same expression as \( f(g(x)) \):\[ g(f(x)) = x^4 + 2x^3 + 2x^2 + x \]

Determine f(g(x)) for b

Given: \( f(x) = \sqrt{x^2 + 2} \) \( g(x) = x^2 \)To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):\[ f(g(x)) = f(x^2) \]Now replace every \( x \) in \( f(x) \) with \( x^2 \):\[ f(x^2) = \sqrt{(x^2)^2 + 2} \]Simplify:\[ \sqrt{x^4 + 2} \]Therefore:\[ f(g(x)) = \sqrt{x^4 + 2} \]

Determine g(f(x)) for b

To find \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \):\[ g(f(x)) = g(\sqrt{x^2 + 2}) \]Now replace every \( x \) in \( g(x) \) with \( \sqrt{x^2 + 2} \):\[ g(\sqrt{x^2 + 2}) = (\sqrt{x^2 + 2})^2 \]Simplify:\[ g(f(x)) = x^2 + 2 \]

Determine f(g(x)) for c

Given: \( f(x) = |x| \) \( g(x) = x^2 \)To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):\[ f(g(x)) = f(x^2) \]Now replace every \( x \) in \( f(x) \) with \( x^2 \):\[ f(x^2) = |x^2| \]Since \( x^2 \) is always non-negative:\[ f(g(x)) = x^2 \]

Determine g(f(x)) for c

To find \( g(f(x)) \), substitute \( f(x) \) into \( g(x) \):\[ g(f(x)) = g(|x|) \]Now replace every \( x \) in \( g(x) \) with \( |x| \):\[ g(|x|) = (|x|)^2 \]Since \( (|x|)^2 = x^2 \):\[ g(f(x)) = x^2 \]

Key concepts

Composite Functions

Composite functions combine two functions into a single function. Let's denote two functions as \( f(x) \) and \( g(x) \). The composite function \( f(g(x)) \) means we first apply \( g(x) \), then use the result as the input for \( f(x) \). The notation \( g(f(x)) \) is another composite function where we first apply \( f(x) \) and then apply \( g(x) \). Composite functions help in understanding how one function modifies the output of another function. They are crucial in mathematics because they let us build more complex functions from simpler ones.
For instance, in the exercise:

• Given: \( f(x) = x^2 + x \) and \( g(x) = x^2 + x \)
  To find \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \):
  \[ f(g(x)) = f(x^2 + x) = (x^2 + x)^2 + (x^2 + x) \]
• Expand and simplify:
  \[ (x^2 + x)^2 = x^4 + 2x^3 + x^2 \]
  So,
  \[ f(g(x)) = x^4 + 2x^3 + 2x^2 + x \]

Understanding and practicing composite functions help you grasp how multiple transformations can be combined.

Precalculus

Precalculus lays the foundation for calculus and includes exploring functions, their properties, and operations. Composite functions are one of the key topics in precalculus. By learning to work with basic to complex functions, you can better understand the continuity, domain, range, and transformations.
In precalculus, functions like \( f(x) = \sqrt{x^2 + 2} \) and their compositions are explored to understand how different inputs are transformed:

• Suppose we need to find \( f(g(x)) \) where \( f(x) = \sqrt{x^2 + 2} \) and \( g(x) = x^2 \).
  First, substitute \( g(x) \) into \( f(x) \):
  \[ f(g(x)) = f(x^2) = \sqrt{(x^2)^2 + 2} \]
• Simplify:
  \[ \sqrt{x^4 + 2} \]
• So,
  \[ f(g(x)) = \sqrt{x^4 + 2} \]

Such exercises develop a deeper conceptual understanding, preparing you for further studies in calculus.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve problems. When working with composite functions, this skill is essential.
Consider the act of substitution and simplification in composite functions. For example:

• Given: \( f(x) = x^2 + x \) and \( g(x) = x^2 + x \), find \( g(f(x)) \).
  Substitute \( f(x) \) into \( g(x) \) and simplify:
  \[ g(f(x)) = g(x^2 + x) = (x^2 + x)^2 + (x^2 + x) \]
• Expanding and simplifying:
  \[ (x^2 + x)^2 = x^4 + 2x^3 + x^2 \]
• Thus:
  \[ g(f(x)) = x^4 + 2x^3 + 2x^2 + x \]

Manipulating algebraic expressions step-by-step helps in obtaining the final simplified form. Practicing these skills ensures you can handle more intricate algebraic processes efficiently.

Function Substitution

Function substitution involves inserting one function into another. This method is crucial for finding compositions such as \( f(g(x)) \) or \( g(f(x)) \). To substitute function \( g \) into \( f \), replace every instance of the variable in \( f \) with \( g(x) \).
Observe the following example:

• Given: \( f(x) = |x| \) and \( g(x) = x^2 \), find \( f(g(x)) \).
  Substitute \( g(x) \) into \( f(x) \):
  \[ f(g(x)) = f(x^2) \]
• Now, replace every \( x \) in \( f(x) \) with \( x^2 \):
  \[ f(x^2) = |x^2| \]
• Since \( x^2 \) is always non-negative:
  \[ f(g(x)) = x^2 \]

Mastering function substitution enables you to dissect complex functions and solve problems by breaking them into manageable steps.