https://file.aiquickdraw.com/mathbot/file/1727438044052ovnn5wzm.jpg

Let's go through each of the function evaluation problems in the image and solve them step by step:

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### **1. Given the function \( f(x) = 3x - 5 \), find \( f(4) \)**:
Substitute \( x = 4 \) into \( f(x) \):
\[
f(4) = 3(4) - 5 = 12 - 5 = 7
\]

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### **2. Find the value of \( h(b) = 3b^2 - 2b + 1 \) when \( b = -3 \)**:
Substitute \( b = -3 \) into \( h(b) \):
\[
h(-3) = 3(-3)^2 - 2(-3) + 1 = 3(9) + 6 + 1 = 27 + 6 + 1 = 34
\]

---

### **3. Find \( g(2w) \) when \( g(x) = x^2 - 2x + 1 \)**:
Substitute \( x = 2w \) into \( g(x) \):
\[
g(2w) = (2w)^2 - 2(2w) + 1 = 4w^2 - 4w + 1
\]

---

### **4. Given \( f(x) = 2x^2 + 4x - 3 \), find \( f(2a + 3) \)**:
Substitute \( x = 2a + 3 \) into \( f(x) \):
\[
f(2a + 3) = 2(2a + 3)^2 + 4(2a + 3) - 3
\]
First, expand \( (2a + 3)^2 \):
\[
(2a + 3)^2 = 4a^2 + 12a + 9
\]
Now, substitute:
\[
f(2a + 3) = 2(4a^2 + 12a + 9) + 4(2a + 3) - 3
\]
\[
= 8a^2 + 24a + 18 + 8a + 12 - 3
\]
\[
= 8a^2 + 32a + 27
\]

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### **5. Given \( f(x) = x^2 - 4 \) and \( g(x) = x + 1 \), evaluate**:
- **a. \( f(-2) \)**:
\[
f(-2) = (-2)^2 - 4 = 4 - 4 = 0
\]
- **b. \( g(3) \)**:
\[
g(3) = 3 + 1 = 4
\]
- **c. \( (f - g)(-1) \)**:
First, evaluate \( f(-1) \) and \( g(-1) \):
\[
f(-1) = (-1)^2 - 4 = 1 - 4 = -3
\]
\[
g(-1) = -1 + 1 = 0
\]
Now, \( (f - g)(-1) = f(-1) - g(-1) = -3 - 0 = -3 \)

- **d. \( f(g(-2)) \)**:
First, find \( g(-2) \):
\[
g(-2) = -2 + 1 = -1
\]
Now, substitute into \( f(x) \):
\[
f(-1) = (-1)^2 - 4 = 1 - 4 = -3
\]

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Let me know if you'd like any of these steps detailed further or have any questions!

### Related Questions:
1. How do you evaluate functions with two variables like \( f(a, b) \)?
2. What happens if the function has a piecewise definition?
3. Can functions involve trigonometric terms, and how would we evaluate those?
4. How do you determine if a function is injective or surjective?
5. What are composite functions, and how are they evaluated?

### Tip:  
When working with composite functions like \( f(g(x)) \), always solve the inside function first, then substitute into the outer function.

Evaluate functions using given formulas for different values of x and b. The problems include finding f(x), h(b), g(2w), and composite functions like f(g(x)).

Learn how to evaluate functions like f(x) = 3x - 5, h(b) = 3b^2 - 2b + 1, and composite functions f(g(x)). This step-by-step guide covers substitution in algebra, helping students in grades 9-12 solve real-world math problems.

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