how to substitutde g ( x ) = 2x + 7 in to f ( x ): 

To substitute \( g(x) = 2x + 7 \) into \( f(x) = x + 3 \), follow these steps:

### Step-by-Step Substitution:

1. **Understand the functions:**
   - \( f(x) = x + 3 \)
   - \( g(x) = 2x + 7 \)

   Here, \( f(x) \) takes an input (which is usually \( x \)) and adds 3 to it. When we substitute \( g(x) \) into \( f(x) \), the input to \( f(x) \) becomes \( g(x) \) instead of just \( x \).

2. **Substitute \( g(x) \) into \( f(x) \):**
   - The notation \( f(g(x)) \) means that wherever you see \( x \) in the function \( f(x) \), you replace it with the expression for \( g(x) \).

   So, starting with \( f(x) = x + 3 \), substitute \( g(x) = 2x + 7 \) in place of \( x \):

   \[
   f(g(x)) = (2x + 7) + 3
   \]

3. **Simplify the expression:**
   - Now, simplify the expression:

   \[
   f(g(x)) = 2x + 7 + 3
   \]

   Combine like terms:

   \[
   f(g(x)) = 2x + 10
   \]

So, the result of substituting \( g(x) = 2x + 7 \) into \( f(x) = x + 3 \) is \( f(g(x)) = 2x + 10 \).

### Recap:
- Start with the function \( f(x) \).
- Replace \( x \) in \( f(x) \) with the expression for \( g(x) \).
- Simplify the result to get the final composed function.

Would you like further clarification or examples?

Here are 5 related questions:
1. How would you substitute a more complex function into another function, like \( g(x) = x^2 + 4x + 1 \) into \( f(x) = 3x - 2 \)?
2. What is the significance of function composition in real-life applications?
3. Can you reverse the process and find \( g(f(x)) \) for these functions?
4. How do you verify if the composed function is correct?
5. What are the conditions for two functions to be composable in terms of their domains?

**Tip:** Always check the domain of the inner function before substitution to ensure the composed function is well-defined.

Learn how to substitute g(x) = 2x + 7 into f(x) = x + 3 with a detailed step-by-step guide. Understand function composition and substitution in algebraic functions. Suitable for students in Grades 10-12.

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