Linear functions are ALWAYS increasing or decreasing but never both.

Question

Quadratic functions will ALWAYS have one interval of increasing and one interval of decreasing. Response area

Exponential functions are always increasing. Response area

true or false

Answer 1

Let's analyze the statements about each type of function:

1. Linear functions: This statement is true. A linear function has a constant slope, meaning it will either always increase (if the slope is positive) or always decrease (if the slope is negative). It cannot be both.

2. Quadratic functions: This statement is true. A standard quadratic function (like \(f(x) = ax^2 + bx + c\), where \(a \neq 0\)) is a parabola. If \(a > 0\), it opens upwards and has one interval of increasing (to the right of the vertex) and one interval of decreasing (to the left of the vertex). If \(a < 0\), it opens downwards and has one interval of decreasing (to the right of the vertex) and one interval of increasing (to the left of the vertex).

3. Exponential functions: This statement is also true. Exponential functions of the form \(f(x) = a \cdot b^x\) (where \(a > 0\) and \(b > 1\)) are always increasing for all \(x\) since the growth rate continuously increases. If \(0 < b < 1\), the function is always decreasing.

In summary, all three statements are true.