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

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0. Functions4h 53m

Introduction to Functions
16m

Piecewise Functions
10m

Properties of Functions
9m

Common Functions
1h 8m

Transformations
5m

Combining Functions
27m

Exponent Rules
32m

Exponential Functions
28m

Logarithmic Functions
24m

Properties of Logarithms
34m

Exponential & Logarithmic Equations
35m

1. Limits and Continuity2h 2m

Introduction to Limits
41m

Finding Limits Algebraically
40m

Continuity
40m

2. Intro to Derivatives1h 33m

Tangent Lines and Derivatives
30m

Derivatives as Functions
14m

Basic Graphing of the Derivative
23m

Differentiability
26m

3. Techniques of Differentiation2h 18m

Basic Rules of Differentiation
43m

Higher Order Derivatives
12m

Product and Quotient Rules
55m

The Chain Rule
27m

4. Derivatives of Exponential & Logarithmic Functions1h 16m

Derivatives of Exponential & Logarithmic Functions
1h 16m

5. Applications of Derivatives2h 19m

Implicit Differentiation
27m

Logarithmic Differentiation
17m

Related Rates
59m

Differentials
21m

Linearization
13m

6. Graphical Applications of Derivatives6h 0m

Intro to Extrema
21m

The First Derivative Test
1h 20m

Concavity
1h 0m

The Second Derivative Test
22m

Curve Sketching
44m

Finding Global Extrema
29m

Applied Optimization
1h 41m

7. Antiderivatives & Indefinite Integrals48m

Antiderivatives
17m

Indefinite Integrals
31m

8. Definite Integrals4h 36m

Estimating Area with Finite Sums
42m

Riemann Sums
1h 16m

Introduction to Definite Integrals
22m

Fundamental Theorem of Calculus
30m

Average Value of a Function
17m

Substitution for Indefinite Integrals
55m

Substitution for Definite Integrals
32m

9. Graphical Applications of Integrals1h 43m

Area Between Curves
1h 19m

Introduction to Volume & Disk Method
23m

10. Integrals of Inverse, Exponential, & Logarithmic Functions21m

Integrals of Exponential Functions
14m

Integrals Involving Logarithmic Functions
7m

11. Techniques of Integration2h 7m

Integration by Parts
1h 37m

Improper Integrals
29m

12. Trigonometric Functions6h 54m

Angles
29m

Trigonometric Functions on Right Triangles
1h 8m

Solving Right Triangles
23m

Trigonometric Functions on the Unit Circle
1h 19m

Graphs of Sine & Cosine
46m

Graphs of Other Trigonometric Functions
32m

Trigonometric Identities
52m

Derivatives of Trig Functions
42m

Integrals of Basic Trig Functions
28m

Integrals of Other Trig Functions
10m

13: Intro to Differential Equations2h 23m

Basics of Differential Equations
43m

Slope Fields
19m

Separable Differential Equations
1h 21m

14. Sequences & Series2h 8m

Sequences
1h 12m

Review of Factorials
11m

Series
44m

15. Power Series2h 19m

Power Series & Taylor Series
2h 19m

16. Probability & Calculus45m

Continuous Probability Models
45m

0. Functions

Properties of Functions

Business Calculus

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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 $open paren 1 comma infinity close paren$

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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.