Question

Example of Function Equation
Linear Function
Quadratic Function
Absolute Value Function
Exponential Growth Function
Exponential Decay Function
Cube Root Function
Cubic Function
Square Root Function

Answer 1

Here are examples of each of the specified function types:

1. **Linear Function**:
\[
f(x) = mx + b
\]
Example:
\[
f(x) = 2x + 3
\]

2. **Quadratic Function**:
\[
f(x) = ax^2 + bx + c
\]
Example:
\[
f(x) = x^2 - 4x + 4
\]

3. **Absolute Value Function**:
\[
f(x) = |x|
\]
Example:
\[
f(x) = |2x - 5|
\]

4. **Exponential Growth Function**:
\[
f(x) = a \cdot b^x \quad \text{(where } b > 1\text{)}
\]
Example:
\[
f(x) = 3 \cdot 2^x
\]

5. **Exponential Decay Function**:
\[
f(x) = a \cdot b^{-x} \quad \text{(or equivalently, } f(x) = a \cdot \frac{1}{b^x} \text{ where } b > 1\text{)}
\]
Example:
\[
f(x) = 5 \cdot (0.5)^x
\]

6. **Cube Root Function**:
\[
f(x) = \sqrt[3]{x}
\]
Example:
\[
f(x) = \sqrt[3]{x - 2}
\]

7. **Cubic Function**:
\[
f(x) = ax^3 + bx^2 + cx + d
\]
Example:
\[
f(x) = 2x^3 - 3x^2 + x - 5
\]

8. **Square Root Function**:
\[
f(x) = \sqrt{x}
\]
Example:
\[
f(x) = \sqrt{x + 3}
\]

Feel free to ask if you need any additional information or clarification on these functions!

Answer 2

what are the math examples

Answer 3

Sure! Here are mathematical examples for each of the mentioned functions, including specific cases for better understanding:

1. **Linear Function**:
- **Function**: \( f(x) = 2x + 3 \)
- **Example Calculation**:
- For \( x = 1 \):
\[
f(1) = 2(1) + 3 = 5
\]
- For \( x = -1 \):
\[
f(-1) = 2(-1) + 3 = 1
\]

2. **Quadratic Function**:
- **Function**: \( f(x) = x^2 - 4x + 4 \)
- **Example Calculation**:
- For \( x = 2 \):
\[
f(2) = 2^2 - 4(2) + 4 = 0
\]
- For \( x = 3 \):
\[
f(3) = 3^2 - 4(3) + 4 = -2
\]

3. **Absolute Value Function**:
- **Function**: \( f(x) = |2x - 5| \)
- **Example Calculation**:
- For \( x = 3 \):
\[
f(3) = |2(3) - 5| = |6 - 5| = 1
\]
- For \( x = 2 \):
\[
f(2) = |2(2) - 5| = |4 - 5| = 1
\]

4. **Exponential Growth Function**:
- **Function**: \( f(x) = 3 \cdot 2^x \)
- **Example Calculation**:
- For \( x = 0 \):
\[
f(0) = 3 \cdot 2^0 = 3 \cdot 1 = 3
\]
- For \( x = 2 \):
\[
f(2) = 3 \cdot 2^2 = 3 \cdot 4 = 12
\]

5. **Exponential Decay Function**:
- **Function**: \( f(x) = 5 \cdot (0.5)^x \)
- **Example Calculation**:
- For \( x = 1 \):
\[
f(1) = 5 \cdot (0.5)^1 = 5 \cdot 0.5 = 2.5
\]
- For \( x = 3 \):
\[
f(3) = 5 \cdot (0.5)^3 = 5 \cdot 0.125 = 0.625
\]

6. **Cube Root Function**:
- **Function**: \( f(x) = \sqrt[3]{x} \)
- **Example Calculation**:
- For \( x = 8 \):
\[
f(8) = \sqrt[3]{8} = 2
\]
- For \( x = -27 \):
\[
f(-27) = \sqrt[3]{-27} = -3
\]

7. **Cubic Function**:
- **Function**: \( f(x) = 2x^3 - 3x^2 + x - 5 \)
- **Example Calculation**:
- For \( x = 1 \):
\[
f(1) = 2(1)^3 - 3(1)^2 + (1) - 5 = 2 - 3 + 1 - 5 = -5
\]
- For \( x = 2 \):
\[
f(2) = 2(2)^3 - 3(2)^2 + (2) - 5 = 16 - 12 + 2 - 5 = 1
\]

8. **Square Root Function**:
- **Function**: \( f(x) = \sqrt{x} \)
- **Example Calculation**:
- For \( x = 9 \):
\[
f(9) = \sqrt{9} = 3
\]
- For \( x = 4 \):
\[
f(4) = \sqrt{4} = 2
\]

These examples illustrate how to compute values for each type of function using provided input values. If you have any further questions or need additional information, feel free to ask!