 f(x)= x-4
g(x)= x^2-8x+16
 solve it in composite function

To solve a composite function problem involving \( f(x) \) and \( g(x) \), we first need to find either \( f(g(x)) \) or \( g(f(x)) \). Here's how we can compute each:

### 1. **Definitions:**
- \( f(x) = x - 4 \)
- \( g(x) = x^2 - 8x + 16 \)

### 2. **Calculating the Composite Functions:**

#### **\( f(g(x)) \):**

1. Substitute \( g(x) \) into \( f \):
\[
f(g(x)) = f(x^2 - 8x + 16)
\]
2. Using the formula for \( f \), where \( f(x) = x - 4 \), replace \( x \) with \( g(x) \):
\[
f(g(x)) = (x^2 - 8x + 16) - 4
\]
3. Simplify the expression:
\[
f(g(x)) = x^2 - 8x + 12
\]

#### **\( g(f(x)) \):**

1. Substitute \( f(x) \) into \( g \):
\[
g(f(x)) = g(x - 4)
\]
2. Using the formula for \( g \), where \( g(x) = x^2 - 8x + 16 \), replace \( x \) with \( f(x) \):
\[
g(f(x)) = (x - 4)^2 - 8(x - 4) + 16
\]
3. Expand the squared term and simplify:
\[
g(f(x)) = (x^2 - 8x + 16) - 8(x - 4) + 16
\]
\[
g(f(x)) = x^2 - 8x + 16 - 8x + 32 + 16
\]
\[
g(f(x)) = x^2 - 16x + 64
\]

### 3. **Summary:**
- \( f(g(x)) = x^2 - 8x + 12 \)
- \( g(f(x)) = x^2 - 16x + 64 \)

Would you like a deeper breakdown of each step or have any questions?

### **Related Questions:**
1. What happens if we evaluate \( f(g(a)) \) for a specific value of \( a \)?
2. Can we find the domain and range of \( f(g(x)) \)?
3. What are the points of intersection between \( f(g(x)) \) and \( g(f(x)) \)?
4. How does the behavior of \( f \) and \( g \) change the composite functions?
5. Can we find the inverse of \( f(g(x)) \)?

### **Tip:**
When finding composite functions, always substitute carefully and simplify step-by-step to avoid errors.

f(x) = x - 4
g(x) = x^2 - 8x + 16
Solve it in composite function

Learn how to solve composite functions involving f(x) = x - 4 and g(x) = x^2 - 8x + 16. This step-by-step guide covers algebraic concepts such as composite functions and quadratic functions, suitable for students in Grades 9-11.

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