91Ó°ÊÓ

Function composition is a way to combine two functions to create a new one. In mathematical terms, if you have two functions, say, \( f(x) \) and \( g(x) \), then the composition \( (f \, \text{∘} \, g)(x) \) is defined as \( f(g(x)) \). This means you first apply function \( g \) to \( x \), and then apply function \( f \) to the result.

Let's illustrate this with the given functions:

• \( f(x) = 3x - 2 \)
• \( g(x) = x^2 + 5 \)

To find \( (f \, \text{∘} \, g)(x) \), we substitute \( g(x) \) into \( f(x) \):
\( (f \, \text{∘} \, g)(x) = f(g(x)) = f(x^2 + 5) \).
We then replace \( x \) in \( f(x) \) with \( x^2 + 5 \):
\( f(x^2 + 5) = 3(x^2 + 5) - 2 \).
Simplifying, we get:
\( 3x^2 + 15 - 2 = 3x^2 + 13 \).
Similarly, to find \( (g \, \text{∘} \, f)(x) \), we substitute \( f(x) \) into \( g(x) \):
\( (g \, \text{∘} \, f)(x) = g(f(x)) = g(3x - 2) \).
Replacing \( x \) in \( g(x) \) with \( 3x - 2 \):
\( g(3x - 2) = (3x - 2)^2 + 5 \).
Simplifying, we get:
\( 9x^2 - 12x + 4 + 5 = 9x^2 - 12x + 9 \).

The domain of a function is the set of all possible inputs (\( x \) values) for which the function is defined.
The domain of the composition of two functions depends on both functions' domains.
For \( f(x) = 3x - 2 \) and \( g(x) = x^2 + 5 \), we need to calculate the domain of both compositions.

Domain of \( (f \, \text{∘} \, g)(x) \):
Since \( g(x) = x^2 + 5 \) is defined for all real numbers, and \( f \) can take any real number as input, the domain of \( (f \, \text{∘} \, g)(x) \) is all real numbers.

Domain of \( (g \, \text{∘} \, f)(x) \):
Here, \( f(x) = 3x - 2 \) is defined for all real numbers; likewise, \( g \) can also take any real number. Thus, the domain of \( (g \, \text{∘} \, f)(x) \) is all real numbers.
So, in both cases, the domain is \( (-\text{∞}, \text{∞}) \).

When working with composed functions, simplifying expressions helps to get the final form:
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• Example for \( (f \, \text{∘} \, g)(x) \):
  Given \( f(x) = 3x - 2 \) and \( g(x) = x^2 + 5 \), we found earlier:
  \( (f \, \text{∘} \, g)(x) = f(x^2 + 5) = 3(x^2 + 5) - 2 \).
  Simplify the expression: \( 3x^2 + 15 - 2 = 3x^2 + 13 \).


• Example for \( (g \, \text{∘} \, f)(x) \):
  Given \( g(x) = x^2 + 5 \) and \( f(x) = 3x - 2 \), we calculated:
  \( (g \, \text{∘} \, f)(x) = g(3x - 2) = (3x - 2)^2 + 5 \).
  Simplify the expression: \( 9x^2 - 12x + 4 + 5 = 9x^2 - 12x + 9 \).

Simplifying helps reduce complex expressions into manageable pieces.

Function evaluation involves finding the output of a function for a specific input.
You replace the variable in the function with the given value and perform the arithmetic.

Example for Composition:

• To evaluate \( (f \, \text{∘} \, g)(x) \) at \( x = 2 \), first find \( g(2) \).
  Given \( g(x) = x^2 + 5 \):
  \( g(2) = 2^2 + 5 = 9 \).
  Next, use the result to find \( f(9) \):
  \( f(x) = 3x - 2 \):
  \( f(9) = 3(9) - 2 = 27 - 2 = 25 \).
  So, \( (f \, \text{∘} \, g)(2) = 25 \).


• For \( (g \, \text{∘} \, f)(x) \) at \( x = 2 \), first find \( f(2) \).
  Given \( f(x) = 3x - 2 \):
  \( f(2) = 3(2) - 2 = 6 - 2 = 4 \).
  Next, use the result to find \( g(4) \):
  \( g(x) = x^2 + 5 \):
  \( g(4) = 4^2 + 5 = 16 + 5 = 21 \).
  So, \( (g \, \text{∘} \, f)(2) = 21 \).

This process helps to understand the value output from composed functions.