https://file.aiquickdraw.com/mathbot/file/1725578389063j7v0ei8y.pdf

The document contains a series of function composition problems. Let's solve them one by one.

### Problem 1:
Given:
\[
f(x) = -4x + 4
\]
\[
g(x) = -x^2 + 4x
\]
Find \( f(g(x)) \).

**Solution:**
\[
f(g(x)) = f(-x^2 + 4x) = -4(-x^2 + 4x) + 4
= 4x^2 - 16x + 4
\]
Thus, \( f(g(x)) = 4x^2 - 16x + 4 \).

---

### Problem 2:
Given:
\[
g(t) = -t - 4
\]
\[
h(t) = t^2 + 4t
\]
Find \( (g \circ h)(t) \), or \( g(h(t)) \).

**Solution:**
\[
g(h(t)) = g(t^2 + 4t) = -(t^2 + 4t) - 4 = -t^2 - 4t - 4
\]
Thus, \( g(h(t)) = -t^2 - 4t - 4 \).

---

### Problem 3:
Given:
\[
f(a) = 4a + 4
\]
\[
g(a) = a^3 - 1 + a
\]
Find \( f(g(a)) \).

**Solution:**
\[
f(g(a)) = f(a^3 - 1 + a) = 4(a^3 - 1 + a) + 4 = 4a^3 + 4a - 4 + 4 = 4a^3 + 4a
\]
Thus, \( f(g(a)) = 4a^3 + 4a \).

---

### Problem 4:
Given:
\[
f(a) = 4a - 3
\]
\[
g(a) = 3a - 4
\]
Find \( f(g(a)) \).

**Solution:**
\[
f(g(a)) = f(3a - 4) = 4(3a - 4) - 3 = 12a - 16 - 3 = 12a - 19
\]
Thus, \( f(g(a)) = 12a - 19 \).

---

### Problem 5:
Given:
\[
g(n) = 4n + 4
\]
\[
h(n) = 2n - 3
\]
Find \( (g \circ h)(n) \), or \( g(h(n)) \).

**Solution:**
\[
g(h(n)) = g(2n - 3) = 4(2n - 3) + 4 = 8n - 12 + 4 = 8n - 8
\]
Thus, \( g(h(n)) = 8n - 8 \).

---

Would you like to see the remaining solutions or have any questions about the process?

Here are 5 related questions to deepen your understanding:
1. How do we compose two functions, and why does the order matter?
2. How can function composition be applied in real-world situations, such as in physics or economics?
3. What are the restrictions on the domain of composed functions?
4. How can the composition of functions be represented graphically?
5. Can we undo the composition of two functions?

**Tip:** When composing functions, always replace the input of the outer function with the entire expression of the inner function.

Perform the indicated operation: f(x) = -4x + 4, g(x) = -x^2 + 4x, Find f(g(x))

Learn how to solve a function composition problem with f(x) = -4x + 4 and g(x) = -x^2 + 4x. This problem demonstrates key algebraic concepts in function composition, suitable for students in Grades 9-11.

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