Given the graph of the following function, determine the intervals on which f(x)f\left(x\right)f(x) is increasing.

( − ∞ , − 2 ) \left(-\infty,-2\right) and ( 1 , ∞ ) \left(1,\infty\right)

Given the graph of the following function, determine the intervals on which f ( x ) f\left(x\right) f ( x ) is increasing.

Given the graph of the following function, determine the intervals on which the function f of x is increasing. Okay. So remember that when we're dealing with increasing functions, the way that we can picture this is by thinking about a little person on our graph here that is walking from left to right. When they're going uphill, that means they're going to be increasing. If they're going downhill, that means decreasing. Now in this case, we're only looking for the situations where we're increasing. Well, I can see we would be going uphill through this portion of the graph, and I can see we'd be going uphill again through that portion of the graph. So if I'm looking for the places where our function is increasing, well, I'm going to see 2 places here. I'm going to see this situation where we go all the way from the left. So it's going to be all the way from negative infinity up to this point right here, which is at negative 2. Now I can see another place that we are increasing is going to be this place right here, which is at 1 all the way to positive infinity. So these are going to be the 2 places on our graph where our function is increasing or where we're going uphill. Now another way that you can think about this is that if you draw these lines that are tangents to your curves, so like a line right there, let's say another line right there, notice that these lines are going to be, in general, trending upward. They're gonna have these upward slopes. However, if I were to draw a line, say, right about here, this would have a downward slope. So you can see because of this that we would not be increasing at that line, but we are increasing at all these places. This is how you can determine where your function is increasing. Let's move on and get some more practice.

Given the graph of the following function, determine the intervals on which f ( x ) f\left(x\right) f ( x ) is increasing.

( − ∞ , − 2 ) \left(-\infty,-2\right) and ( 1 , ∞ ) \left(1,\infty\right)

( − 4 , − 3 ) \left(-4,-3\right) ( − 4 , − 3 ) and ( 2 , 4 ) \left(2,4\right) ( 2 , 4 )

[ − 4 , − 2 ] \left\lbrack-4,-2\right\rbrack and [ 1 , 4 ] \left\lbrack1,4\right\rbrack

[ − 2 , − 1 ] \left\lbrack-2,-1\right\rbrack

0. Functions

Properties of Functions

Business Calculus

0. Functions

Properties of Functions

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Multiple Choice

Given the graph of the following function, determine the intervals on which $f\left(x\right)$ is increasing.

$\left(-4,-3\right)$ and $\left(2,4\right)$

$[- 4, - 2]$ and $open bracket 1 comma 4 close bracket$

$[- 2, - 1]$

$(- \infty, - 2)$ and $open paren 1 comma infinity close paren$

Verified step by step guidance

Step 1: Analyze the graph of the function f(x) to identify where the slope of the curve is positive. A positive slope indicates that the function is increasing.

Step 2: Observe the graph and note that the function decreases from left to right until x = -2. After x = -2, the slope becomes positive, indicating the function starts increasing.

Step 3: Continue analyzing the graph and observe that the function increases until x = 1. After x = 1, the slope remains positive, indicating the function continues to increase.

Step 4: Based on the observations, the intervals where f(x) is increasing are (-∞, -2) and (1, ∞). These intervals correspond to the regions where the slope of the graph is positive.

Step 5: Verify the intervals by checking the behavior of the graph visually and ensuring that the function does not decrease within these intervals.

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