Answer :
To solve this problem, let's analyze the different types of functions listed and determine if their graphs have intervals that both increase and decrease:
1. Absolute Value Function:
The graph of an absolute value function, like [tex]\( f(x) = |x| \)[/tex], is V-shaped. This means that there is a portion of the graph that increases (e.g., for [tex]\( x > 0 \)[/tex]) and another portion that decreases (e.g., for [tex]\( x < 0 \)[/tex]). Thus, the absolute value function has intervals where it increases and decreases.
2. Quadratic Function:
The graph of a quadratic function, like [tex]\( f(x) = ax^2 + bx + c \)[/tex] where [tex]\( a \neq 0 \)[/tex], is a parabola. If the parabola opens upwards ([tex]\( a > 0 \)[/tex]), the function decreases on the interval to the left of its vertex and increases on the interval to the right of its vertex. If the parabola opens downwards ([tex]\( a < 0 \)[/tex]), the function increases on the interval to the left of its vertex and decreases on the interval to the right of its vertex. Therefore, a quadratic function has intervals that are increasing and decreasing.
3. None of the Above:
This option is not suitable since we have identified at least two types of functions (absolute value and quadratic) that meet the criteria.
4. Linear Function:
The graph of a linear function, like [tex]\( f(x) = mx + b \)[/tex], is a straight line. Depending on the slope ([tex]\( m \)[/tex]), it can only either increase or decrease but not both. If [tex]\( m > 0 \)[/tex], it increases; if [tex]\( m < 0 \)[/tex], it decreases. Hence, a linear function does not have intervals that both increase and decrease.
Based on this analysis, the functions that have graphs with at least one interval that increases and one interval that decreases are:
- Absolute value function
- Quadratic function
The correct options are the first and second choices. Thus, the answer is:
[tex]\[ \boxed{[1, 2]} \][/tex]
1. Absolute Value Function:
The graph of an absolute value function, like [tex]\( f(x) = |x| \)[/tex], is V-shaped. This means that there is a portion of the graph that increases (e.g., for [tex]\( x > 0 \)[/tex]) and another portion that decreases (e.g., for [tex]\( x < 0 \)[/tex]). Thus, the absolute value function has intervals where it increases and decreases.
2. Quadratic Function:
The graph of a quadratic function, like [tex]\( f(x) = ax^2 + bx + c \)[/tex] where [tex]\( a \neq 0 \)[/tex], is a parabola. If the parabola opens upwards ([tex]\( a > 0 \)[/tex]), the function decreases on the interval to the left of its vertex and increases on the interval to the right of its vertex. If the parabola opens downwards ([tex]\( a < 0 \)[/tex]), the function increases on the interval to the left of its vertex and decreases on the interval to the right of its vertex. Therefore, a quadratic function has intervals that are increasing and decreasing.
3. None of the Above:
This option is not suitable since we have identified at least two types of functions (absolute value and quadratic) that meet the criteria.
4. Linear Function:
The graph of a linear function, like [tex]\( f(x) = mx + b \)[/tex], is a straight line. Depending on the slope ([tex]\( m \)[/tex]), it can only either increase or decrease but not both. If [tex]\( m > 0 \)[/tex], it increases; if [tex]\( m < 0 \)[/tex], it decreases. Hence, a linear function does not have intervals that both increase and decrease.
Based on this analysis, the functions that have graphs with at least one interval that increases and one interval that decreases are:
- Absolute value function
- Quadratic function
The correct options are the first and second choices. Thus, the answer is:
[tex]\[ \boxed{[1, 2]} \][/tex]
