GPT4

To make a table for h(x)=g(x−1)+5*h*(*x*)=*g*(*x*−1)+5 for values of x=1,2,3,4*x*=1,2,3,4, you'll need to follow these steps:

To make a table for the function \( h(x) = g(x-1) + 5 \) for \( x = 1, 2, 3, 4 \), follow these steps:

### Steps:

1. **Understand the function**: 
   The given function is \( h(x) = g(x-1) + 5 \). This means that for any value of \( x \), \( h(x) \) depends on the value of \( g(x-1) \), with an additional constant term of 5.

2. **Define or obtain \( g(x) \)**: 
   The function \( g(x) \) must be defined or given to calculate \( g(x-1) \). If the values of \( g(x) \) are known, substitute them into \( g(x-1) \) for each \( x \). Without specific information on \( g(x) \), we can only express the results in terms of \( g \).

3. **Substitute values of \( x \)**: 
   Now, substitute the values of \( x = 1, 2, 3, 4 \) into the formula to compute \( h(x) \).

### Substitution:

- For \( x = 1 \): 
  \[
  h(1) = g(1-1) + 5 = g(0) + 5
  \]
- For \( x = 2 \): 
  \[
  h(2) = g(2-1) + 5 = g(1) + 5
  \]
- For \( x = 3 \): 
  \[
  h(3) = g(3-1) + 5 = g(2) + 5
  \]
- For \( x = 4 \): 
  \[
  h(4) = g(4-1) + 5 = g(3) + 5
  \]

### Table Structure:

| \( x \) | \( g(x-1) \) | \( h(x) = g(x-1) + 5 \) |
|--------|--------------|-------------------------|
| 1      | \( g(0) \)   | \( g(0) + 5 \)          |
| 2      | \( g(1) \)   | \( g(1) + 5 \)          |
| 3      | \( g(2) \)   | \( g(2) + 5 \)          |
| 4      | \( g(3) \)   | \( g(3) + 5 \)          |

### Final Notes:
- You need to know the values of \( g(x) \) to complete the table. Without knowing \( g(x) \), the result stays expressed in terms of \( g \).

Would you like to provide specific values for \( g(x) \) to fill in the table? Or would you like more details on any of the steps?

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Here are 5 related questions you might explore:

1. What happens if \( g(x) \) is a specific polynomial function, like \( g(x) = 2x + 1 \)?
2. How does the table change if \( h(x) \) includes other constants besides 5?
3. What is the interpretation of \( g(x-1) \) in practical problems?
4. How can we differentiate or integrate \( h(x) \) if \( g(x) \) is continuous?
5. How do different transformations of \( g(x) \) affect the form of \( h(x) \)?

**Tip**: Always clarify the domain and range of the function \( g(x) \) before working with composite functions like \( h(x) \).

To make a table for h(x)=g(x−1)+5 for values of x=1,2,3,4, you'll need to follow these steps:

This guide walks you through creating a table for the function h(x) = g(x-1) + 5 for values of x = 1, 2, 3, 4. It covers key algebraic concepts such as function composition and substitution. Suitable for students in Grades 9-12.

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