B. Evaluate the following exponential functions. (10 points)

1. f(x) = 2 ^ x, f(- 3) 

2. f(x) = (1/3) ^ x, f(- 2)

3. 3. f(x) = 4 ^ x, f(- 2)

4. f(x) = 5 ^ x, f(- 1)

5. f(x) = (2/3) ^ x, f(2) 

6. f(x) = (4/3) ^ x, f(2)

7. f(x) = 6 ^ x, f(- 1)

8. f(x) = 5 ^ - x, f(3)

9. f(x) = (1/7) ^ x, f(1)

10. f(x) = (3/4) ^ x, f(- 3)

Let's evaluate the given exponential functions one by one:

### 1. \( f(x) = 2^x \), find \( f(-3) \):
\[
f(-3) = 2^{-3} = \frac{1}{2^3} = \frac{1}{8}
\]
So, \( f(-3) = \frac{1}{8} \).

---

### 2. \( f(x) = \left(\frac{1}{3}\right)^x \), find \( f(-2) \):
\[
f(-2) = \left(\frac{1}{3}\right)^{-2} = \left(\frac{3}{1}\right)^2 = 3^2 = 9
\]
So, \( f(-2) = 9 \).

---

### 3. \( f(x) = 4^x \), find \( f(-2) \):
\[
f(-2) = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}
\]
So, \( f(-2) = \frac{1}{16} \).

---

### 4. \( f(x) = 5^x \), find \( f(-1) \):
\[
f(-1) = 5^{-1} = \frac{1}{5}
\]
So, \( f(-1) = \frac{1}{5} \).

---

### 5. \( f(x) = \left(\frac{2}{3}\right)^x \), find \( f(2) \):
\[
f(2) = \left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}
\]
So, \( f(2) = \frac{4}{9} \).

---

### 6. \( f(x) = \left(\frac{4}{3}\right)^x \), find \( f(2) \):
\[
f(2) = \left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9}
\]
So, \( f(2) = \frac{16}{9} \).

---

### 7. \( f(x) = 6^x \), find \( f(-1) \):
\[
f(-1) = 6^{-1} = \frac{1}{6}
\]
So, \( f(-1) = \frac{1}{6} \).

---

### 8. \( f(x) = 5^{-x} \), find \( f(3) \):
\[
f(3) = 5^{-3} = \frac{1}{5^3} = \frac{1}{125}
\]
So, \( f(3) = \frac{1}{125} \).

---

### 9. \( f(x) = \left(\frac{1}{7}\right)^x \), find \( f(1) \):
\[
f(1) = \left(\frac{1}{7}\right)^1 = \frac{1}{7}
\]
So, \( f(1) = \frac{1}{7} \).

---

### 10. \( f(x) = \left(\frac{3}{4}\right)^x \), find \( f(-3) \):
\[
f(-3) = \left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3 = \frac{4^3}{3^3} = \frac{64}{27}
\]
So, \( f(-3) = \frac{64}{27} \).

---

### Summary of Results:
1. \( f(-3) = \frac{1}{8} \)
2. \( f(-2) = 9 \)
3. \( f(-2) = \frac{1}{16} \)
4. \( f(-1) = \frac{1}{5} \)
5. \( f(2) = \frac{4}{9} \)
6. \( f(2) = \frac{16}{9} \)
7. \( f(-1) = \frac{1}{6} \)
8. \( f(3) = \frac{1}{125} \)
9. \( f(1) = \frac{1}{7} \)
10. \( f(-3) = \frac{64}{27} \)

Would you like more details or have any further questions?

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Here are 5 related questions to explore:
1. What is the general formula for evaluating \( f(x) = a^x \) for negative values of \( x \)?
2. How does changing the base of an exponential function affect the graph of the function?
3. What happens when the base of an exponential function is between 0 and 1?
4. Can an exponential function ever equal zero? Why or why not?
5. What is the inverse of an exponential function?

**Tip:** When dealing with negative exponents, remember that \( a^{-x} = \frac{1}{a^x} \).

Evaluate the following exponential functions: 1. f(x) = 2^x, f(-3), 2. f(x) = (1/3)^x, f(-2), 3. f(x) = 4^x, f(-2), 4. f(x) = 5^x, f(-1), 5. f(x) = (2/3)^x, f(2), 6. f(x) = (4/3)^x, f(2), 7. f(x) = 6^x, f(-1), 8. f(x) = 5^-x, f(3), 9. f(x) = (1/7)^x, f(1), 10. f(x) = (3/4)^x, f(-3).

Learn how to evaluate exponential functions with negative exponents and fractional bases. This example problem walks through evaluating functions like f(x) = 2^x and f(x) = (3/4)^x for negative and positive values of x. Suitable for students in Grades 9-10.

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