Evaluating Exponential Functions Worksheets - Free Printable

Problem: Evaluating Exponential Functions

We will solve each part of the problem step by step.

---

Part A: Evaluate each function at the specified value

#### 1) \( f(x) = 9^{(x-2)} \); \( x = 10 \)

Substitute \( x = 10 \) into the function:
\[
f(10) = 9^{(10-2)} = 9^8
\]

Now, calculate \( 9^8 \):
\[
9^8 = (3^2)^8 = 3^{16}
\]
Using a calculator or direct computation:
\[
3^{16} = 43046721
\]

Thus, the answer is:
\[
\boxed{43046721}
\]

---

#### 2) \( f(x) = 8 \cdot (-2)^{(x-3)} - 3x \); \( x = 2 \)

Substitute \( x = 2 \) into the function:
\[
f(2) = 8 \cdot (-2)^{(2-3)} - 3(2)
\]

Simplify the exponent:
\[
f(2) = 8 \cdot (-2)^{-1} - 6
\]

Recall that \( (-2)^{-1} = \frac{1}{-2} = -\frac{1}{2} \):
\[
f(2) = 8 \cdot \left(-\frac{1}{2}\right) - 6
\]

Multiply:
\[
f(2) = -4 - 6 = -10
\]

Thus, the answer is:
\[
\boxed{-10}
\]

---

#### 3) \( f(x) = -12 + 12^{(2x + \frac{x}{2})} \); \( x = -4 \)

Substitute \( x = -4 \) into the function:
\[
f(-4) = -12 + 12^{(2(-4) + \frac{-4}{2})}
\]

Simplify the exponent:
\[
2(-4) + \frac{-4}{2} = -8 + (-2) = -10
\]

So:
\[
f(-4) = -12 + 12^{-10}
\]

Since \( 12^{-10} \) is a very small number (close to 0), we can approximate:
\[
f(-4) \approx -12
\]

Thus, the answer is:
\[
\boxed{-12}
\]

---

#### 4) \( f(x) = x - 10^{-2x} \); \( x = -1 \)

Substitute \( x = -1 \) into the function:
\[
f(-1) = (-1) - 10^{-2(-1)}
\]

Simplify the exponent:
\[
-2(-1) = 2
\]

So:
\[
f(-1) = -1 - 10^2
\]

Calculate \( 10^2 \):
\[
10^2 = 100
\]

Thus:
\[
f(-1) = -1 - 100 = -101
\]

The answer is:
\[
\boxed{-101}
\]

---

Part B: Evaluate each function

#### 1) If \( f(x) = 10 \cdot 4^x - 11 \), find \( f(-2) \).

Substitute \( x = -2 \) into the function:
\[
f(-2) = 10 \cdot 4^{-2} - 11
\]

Recall that \( 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \):
\[
f(-2) = 10 \cdot \frac{1}{16} - 11
\]

Simplify:
\[
f(-2) = \frac{10}{16} - 11 = \frac{5}{8} - 11
\]

Convert 11 to a fraction with denominator 8:
\[
11 = \frac{88}{8}
\]

So:
\[
f(-2) = \frac{5}{8} - \frac{88}{8} = \frac{5 - 88}{8} = \frac{-83}{8}
\]

The answer is:
\[
\boxed{-\frac{83}{8}}
\]

---

#### 2) If \( f(x) = 4^{(x-2)} + 1 \), find \( f(3) \).

Substitute \( x = 3 \) into the function:
\[
f(3) = 4^{(3-2)} + 1
\]

Simplify the exponent:
\[
3 - 2 = 1
\]

So:
\[
f(3) = 4^1 + 1 = 4 + 1 = 5
\]

The answer is:
\[
\boxed{5}
\]

---

#### 3) If \( f(x) = 13 \cdot (-5)^{(x+2)} \), find \( f(-8) \).

Substitute \( x = -8 \) into the function:
\[
f(-8) = 13 \cdot (-5)^{(-8+2)}
\]

Simplify the exponent:
\[
-8 + 2 = -6
\]

So:
\[
f(-8) = 13 \cdot (-5)^{-6}
\]

Recall that \( (-5)^{-6} = \frac{1}{(-5)^6} \):
\[
(-5)^6 = 5^6 = 15625
\]

Thus:
\[
(-5)^{-6} = \frac{1}{15625}
\]

So:
\[
f(-8) = 13 \cdot \frac{1}{15625} = \frac{13}{15625}
\]

The answer is:
\[
\boxed{\frac{13}{15625}}
\]

---

#### 4) If \( f(x) = -7 + (-1)^{(x-1)} \), find \( f(-5) \).

Substitute \( x = -5 \) into the function:
\[
f(-5) = -7 + (-1)^{(-5-1)}
\]

Simplify the exponent:
\[
-5 - 1 = -6
\]

So:
\[
f(-5) = -7 + (-1)^{-6}
\]

Recall that \( (-1)^{-6} = \frac{1}{(-1)^6} \). Since \( (-1)^6 = 1 \):
\[
(-1)^{-6} = 1
\]

Thus:
\[
f(-5) = -7 + 1 = -6
\]

The answer is:
\[
\boxed{-6}
\]

---

Part C: What is the value of \( f(0) \) if \( f(x) = (-6)^{(2-x)} + x \)?

Substitute \( x = 0 \) into the function:
\[
f(0) = (-6)^{(2-0)} + 0
\]

Simplify the exponent:
\[
2 - 0 = 2
\]

So:
\[
f(0) = (-6)^2 + 0
\]

Calculate \( (-6)^2 \):
\[
(-6)^2 = 36
\]

Thus:
\[
f(0) = 36
\]

The answer is:
\[
\boxed{36}
\]

---

Final Answers:

\[
\boxed{43046721, -10, -12, -101, -\frac{83}{8}, 5, \frac{13}{15625}, -6, 36}
\]