Question

determine all intervals on which the graph of ( f ) is increasing.

Explanation:

Step1: Recall the definition of an increasing function

A function \( f(x) \) is increasing on an interval if, as \( x \) increases (moves from left to right along the x - axis), the value of \( f(x) \) (the y - value) also increases. Geometrically, this means the graph of the function rises as we move from left to right over that interval.

Step2: Analyze the graph

• First, look at the left - most part of the graph. We can see that when \( x \) is in the interval \( (-\infty, - 3) \) (assuming the peak of the left - hand curve is at \( x=-3 \)), as \( x \) increases from \( -\infty \) to \( - 3 \), the \( y \) - values of the function are increasing (the graph is rising from the bottom left towards the peak at \( x = - 3 \)).
• Then, look at the right - hand part of the graph (the part near the positive x - axis). We can see that when \( x \) is in the interval \( (1, 5) \) (assuming the minimum on the right - hand side is at \( x = 1 \) and the peak is at \( x=5 \)), as \( x \) increases from \( 1 \) to \( 5 \), the \( y \) - values of the function are increasing (the graph is rising from the minimum at \( x = 1 \) towards the peak at \( x = 5 \)).

Answer:

\((-\infty, - 3)\cup(1, 5)\)