Consider the functions f(x)=2x2−3 and g(x)=−x+6.

f(g(4))=

g(f(4))=

Let's solve both of these expressions step by step.

### 1. **Finding \( f(g(4)) \):**

- First, we need to find \( g(4) \):
  \[
  g(x) = -x + 6
  \]
  Substitute \( x = 4 \) into \( g(x) \):
  \[
  g(4) = -(4) + 6 = -4 + 6 = 2
  \]
- Now, we use this result to find \( f(g(4)) = f(2) \):
  \[
  f(x) = 2x^2 - 3
  \]
  Substitute \( x = 2 \) into \( f(x) \):
  \[
  f(2) = 2(2)^2 - 3 = 2(4) - 3 = 8 - 3 = 5
  \]
Thus, \( f(g(4)) = 5 \).

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### 2. **Finding \( g(f(4)) \):**

- First, we need to find \( f(4) \):
  \[
  f(x) = 2x^2 - 3
  \]
  Substitute \( x = 4 \) into \( f(x) \):
  \[
  f(4) = 2(4)^2 - 3 = 2(16) - 3 = 32 - 3 = 29
  \]
- Now, we use this result to find \( g(f(4)) = g(29) \):
  \[
  g(x) = -x + 6
  \]
  Substitute \( x = 29 \) into \( g(x) \):
  \[
  g(29) = -(29) + 6 = -29 + 6 = -23
  \]
Thus, \( g(f(4)) = -23 \).

### Final Answers:
- \( f(g(4)) = 5 \)
- \( g(f(4)) = -23 \)

Do you want more details or have any questions?

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Here are 5 related questions:
1. What is the general form of \( f(g(x)) \) for these functions?
2. How would \( f(g(x)) \) change if \( g(x) = -2x + 5 \)?
3. What is \( f(f(4)) \)?
4. How would the results change if \( f(x) = x^2 - 3x \)?
5. Can we solve for \( g(g(4)) \)?

**Tip**: Always solve functions step by step, ensuring substitution is done accurately for nested functions.

Consider the functions f(x)=2x^2−3 and g(x)=−x+6.
f(g(4))=
g(f(4))=

Explore step-by-step solutions for the function composition problem with f(x)=2x^2−3 and g(x)=−x+6. Learn how to compute f(g(4)) and g(f(4)) using substitution, quadratic, and linear functions. Suitable for students in Grades 9-11.

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