Question

What equations represent functions? Assume x is the input and y is the output.

Answer 1

A function is a relation that assigns exactly one output (y) for each input (x). Here are some examples of equations that represent functions:

1. **Linear Function**:
- \( y = mx + b \)
- Example: \( y = 2x + 3 \)

2. **Quadratic Function**:
- \( y = ax^2 + bx + c \)
- Example: \( y = x^2 - 5x + 6 \)

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

4. **Exponential Function**:
- \( y = ab^x \)
- Example: \( y = 2 \cdot 3^x \)

5. **Logarithmic Function**:
- \( y = a \log_b(x) + c \)
- Example: \( y = \log_2(x) + 1 \)

6. **Rational Function**:
- \( y = \frac{p(x)}{q(x)} \) where \( q(x) \neq 0 \)
- Example: \( y = \frac{x^2 - 1}{x + 1} \)

7. **Trigonometric Function**:
- \( y = a \sin(bx + c) + d \)
- Example: \( y = 3 \sin(x) + 2 \)

8. **Absolute Value Function**:
- \( y = |x| \)
- Example: \( y = |x - 2| \)

9. **Piecewise Function**:
- Defined by different expressions for different intervals of x.
- Example:
\[
y =
\begin{cases}
x^2 & \text{if } x < 0 \\
x + 1 & \text{if } x \geq 0
\end{cases}
\]

In contrast, equations that do not represent functions include those that assign multiple outputs to a single input, such as:

- \( y^2 = x \) (as for some values of x, there are two y-values).
- \( y = \sqrt{x} - 1 \) (is a function, but if we replace with \( y^4 = x \), it would no longer be).

To determine whether a relation is a function, you can also use the **vertical line test**: if any vertical line intersects the graph of the relation at more than one point, it is not a function.