Question

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

answer attempt 1 out of 2
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symbols: (, ), , , {, }, <, >, ≤, ≥, ∞, all real numbers

Explanation:

Step1: Recall Increasing Function Definition

A function \( f(x) \) is increasing on an interval if, as \( x \) increases (moves from left to right along the x - axis), \( f(x) \) also increases (the graph rises).

Step2: Analyze the Graph

- First, find the critical points (where the graph changes from increasing to decreasing or vice - versa) by looking at the peaks and valleys of the graph.
- From the graph, we can see that the function has a minimum (valley) at \( x=-3 \) (approximately, looking at the grid) and then another minimum (valley) and a maximum (peak) in the positive x - region. But to find where it's increasing, we look at the intervals where the slope of the tangent line is positive (the graph is rising).
- Starting from the leftmost part: As we move from \( x = - \infty \) towards \( x=-3 \), the graph is decreasing (going down). Then, from \( x=-3 \) onwards, when we move to the right, the graph starts to rise until it reaches a peak, then it falls, and then rises again. Wait, actually, let's re - examine. Wait, the left - most part: when \( x\) increases from \( - \infty \) to \( - 3\), the graph is decreasing (since it goes from a higher y - value to a lower y - value at \( x = - 3\)). Then, from \( x=-3\) to the peak (let's say around \( x = 4\) or so, but more accurately, looking at the graph, the first interval where it's increasing is from \( x=-3\) to the peak, and then after the valley (around \( x = 7\) or so), it's increasing again from \( x = 7\) to \( \infty\)? Wait, no, maybe I misread. Wait, the graph: let's see the key points. The function has a minimum at \( x=-3\) (the lowest point on the left - hand side valley), then it increases until a peak, then decreases until a valley, then increases again. So the intervals where it's increasing are:
- From \( x=-3\) to the x - value of the first peak (let's assume the first peak is at \( x = 4\), but actually, looking at the grid, the left valley is at \( x=-3\) (since the graph goes down to \( x=-3\) and then up), then it goes up until a peak, then down, then up again. Wait, maybe the correct intervals are \( (-3, 4)\) and \( (7, \infty)\)? Wait, no, let's do it properly. The definition of an increasing function on an interval \( (a,b) \) is that for any \( x_1,x_2\in(a,b) \) with \( x_1<x_2 \), \( f(x_1)<f(x_2) \). So we look at the graph's slope. When the graph is rising (going up as we move from left to right), the function is increasing.
- The first interval: starting from the minimum at \( x = - 3\) (the left - hand valley), as we move to the right, the graph rises until it reaches a peak (let's say at \( x = 4\)). Then it falls until it reaches a valley (at \( x = 7\)), then it rises again as we move to the right towards \( \infty \).
- So the intervals where \( f(x) \) is increasing are \( (-3, 4) \) and \( (7, \infty) \). Wait, but maybe the graph is such that the first increasing interval is from \( x=-3\) to \( \infty \) with a dip? No, the graph has a peak and then a valley. Wait, maybe the correct way is to look at the critical points. Let's assume the critical points (where the derivative is zero, i.e., local maxima and minima) are at \( x=-3\), \( x = 4\), and \( x = 7\). Then:
- For \( x < - 3\): the function is decreasing (since as \( x\) increases towards \( - 3\), \( f(x)\) decreases).
- For \( - 3<x<4\): the function is increasing (as \( x\) increases, \( f(x)\) increases).
- For \( 4<x<7\): the function is decreasing (as \( x\) increases, \( f(x)\) decreases).
- For \( x > 7\): the function is increasing (as \( x\) increases, \( f(x)\) increases).

Answer:

The intervals on which the graph of \( f \) is increasing are \( (-3, 4) \) and \( (7, \infty) \) (Note: The actual intervals may vary slightly depending on the exact coordinates of the critical points from the graph, but the general method is as above. If the graph's critical points are at \( x=-3\), \( x = 4\), and \( x = 7\), then the increasing intervals are \( (-3, 4)\) and \( (7, \infty) \))