example of a composite function being represented numerically

To represent a composite function numerically, let's consider two functions \( f(x) \) and \( g(x) \). The composite function \( (f \circ g)(x) \) means applying \( g(x) \) first, then applying \( f \) to the result of \( g(x) \).

### Example:

Let:
- \( f(x) = 2x + 3 \)
- \( g(x) = x^2 - 1 \)

The composite function \( (f \circ g)(x) \) is defined as:
\[
(f \circ g)(x) = f(g(x))
\]

**Step-by-step Calculation:**

1. **Find \( g(x) \):**
   - For \( x = 1 \): \( g(1) = 1^2 - 1 = 0 \)
   - For \( x = 2 \): \( g(2) = 2^2 - 1 = 3 \)
   - For \( x = 3 \): \( g(3) = 3^2 - 1 = 8 \)

2. **Apply \( f \) to the values of \( g(x) \):**
   - For \( g(1) = 0 \): \( f(g(1)) = f(0) = 2(0) + 3 = 3 \)
   - For \( g(2) = 3 \): \( f(g(2)) = f(3) = 2(3) + 3 = 9 \)
   - For \( g(3) = 8 \): \( f(g(3)) = f(8) = 2(8) + 3 = 19 \)

**Numerical Representation of \( (f \circ g)(x) \):**
- At \( x = 1 \), \( (f \circ g)(1) = 3 \)
- At \( x = 2 \), \( (f \circ g)(2) = 9 \)
- At \( x = 3 \), \( (f \circ g)(3) = 19 \)

So, the values of the composite function \( (f \circ g)(x) \) for \( x = 1, 2, 3 \) are 3, 9, and 19, respectively.

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Would you like more details on this, or do you have any questions?

Here are some related questions to explore further:

1. How do you graph a composite function?
2. What happens when you switch the order of composition, like \( (g \circ f)(x) \)?
3. How do you find the domain of a composite function?
4. How do you decompose a composite function into its component functions?
5. Can a composite function be simplified algebraically?

**Tip:** When evaluating composite functions, always remember to apply the inner function first, then the outer function to the result.

Explore how to represent a composite function numerically with detailed steps. This example uses f(x) = 2x + 3 and g(x) = x^2 - 1, demonstrating the composition of functions for x = 1, 2, and 3. Suitable for students in Grades 9-12.

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