Use the given graph of the function f to answer parts...

Use the given graph of the function $f$ to answer parts (a)-(n). (GRAPH CANNOT COPY) (a) Find $f(0)$ and $f(6)$ (b) Find $f(2)$ and $f(-2)$ (c) Is $f(3)$ positive or negative? (d) Is $f(-1)$ positive or negative? (e) For what values of $x$ is $f(x)=0 ?$ (f) For what values of $x$ is $f(x)<0 ?$ (g) What is the domain of $f ?$ (h) What is the range of $f ?$ (i) What are the $x$ -intercepts? (j) What is the $y$ -intercept? (k) How often does the line $y=-1$ intersect the graph? (1) How often does the line $x=1$ intersect the graph? (m) For what value of $x$ does $f(x)=3 ?$ (n) For what value of $x$ does $f(x)=-2 ?$

Use the given graph of the function $f$ to answer parts (a)-(n).
(GRAPH CANNOT COPY)
(a) Find $f(0)$ and $f(6)$
(b) Find $f(2)$ and $f(-2)$
(c) Is $f(3)$ positive or negative?
(d) Is $f(-1)$ positive or negative?
(e) For what values of $x$ is $f(x)=0 ?$
(f) For what values of $x$ is $f(x)<0 ?$
(g) What is the domain of $f ?$
(h) What is the range of $f ?$
(i) What are the $x$ -intercepts?
(j) What is the $y$ -intercept?
(k) How often does the line $y=-1$ intersect the graph?
(1) How often does the line $x=1$ intersect the graph?
(m) For what value of $x$ does $f(x)=3 ?$
(n) For what value of $x$ does $f(x)=-2 ?$

Key Concepts

Graphical Inequality Analysis

This concept involves determining where the function’s graph lies above or below a reference line, typically the x-axis. For instance, identifying intervals where f(x) is negative or positive requires observing whether the graph is below or above the x-axis respectively. This skill is utilized when answering questions about the positivity or negativity of function values.

Intercepts

Intercepts of a graph are the points where it crosses the axes. The x-intercepts are the points where f(x)=0 (the zeros), and the y-intercept is the point where the graph crosses the y-axis (f(0)). Recognizing and reading these intercepts from the graph is important in analyzing overall function behavior.

Intersection with Horizontal and Vertical Lines

Analyzing intersections between the function’s graph and lines like y = c (horizontal) or x = c (vertical) is a valuable technique. It helps to find the number of solutions to equations such as f(x)=c by counting the crossing points. This approach is integral when a problem asks how many times a line, like y = -1 or x = 1, intersects the function.

Zeros of a Function

Zeros are the x-values for which the function equals zero (f(x)=0). Graphically, these correspond to the points where the graph crosses or touches the x-axis. This concept is key when a problem asks for the solutions of f(x)=0 or the x-intercepts of the graph.

Graphical Function Evaluation

This concept involves interpreting a function’s value at a specific input directly from its graph. By locating the x-coordinate on the horizontal axis and finding the corresponding y-coordinate on the graph, you obtain f(x). This process is fundamental for answering questions that ask for specific function outputs, such as finding f(0) or f(6).

Domain and Range

The domain represents all possible x-values for which the function is defined, while the range consists of all y-values that the function attains. Determining these from a graph involves examining the extent of the graph horizontally and vertically. Understanding these concepts is critical in questions about the set of inputs and outputs of the function.

Transcript

00:01 Using the graph above, we're going to answer a whole bunch of questions that we have.

00:05 So first is find f of zero and f of six.

00:09 So remember that this means that x is zero.

00:13 And here, this means x is six.

00:15 So what we want to do is we want to look at our x axis.

00:19 And right here, this is where x is zero.

00:22 Well, the coordinate, the y part of that is zero.

00:26 So f of zero equals zero.

00:30 Now let's look along our x -axis and we're going to see here is six.

00:35 Well, the y -cordinate when x is six is zero, so f -of -6 equals zero as well.

00:43 Now, let's take a look at the next one.

00:46 We're going to find f -of -2 and negative two.

00:48 We're going to do the same thing.

00:51 So if you look, here is where x is two, but there's no point there.

00:56 There's no graph there.

00:57 So if we go down, now we see the point, and the y coordinate is negative 2.

01:03 So when x is 2, y is negative 2.

01:09 Let's take a look at when x is negative 2.

01:12 So i'm going to go along the x -axis to negative 2, and then i have to go up and here is my ordered pair.

01:18 There is a point on the graph.

01:20 So when f of negative 2 is positive 1, because that's where the point is on the graph.

01:30 Now, let me raise some of this.

01:32 Let's take a look at f of 3.

01:35 Here's f of 3, which is between 2 and 4, and now when i go down, it's somewhere here on the graph.

01:44 It's just asking, is f of 3 positive or negative? well, since it's below the x -axis, it has to be negative.

01:54 We're not asking for the exact number.

01:56 They just want to know if it would be a negative number or positive number.

01:59 Now let's do the same thing.

02:02 If f is f of negative 1, is f of negative 1 positive or negative? so let's go over to negative 1.

02:10 And if i go up to the graph, it is above the x -axis.

02:14 So it has to be positive.

02:18 For all the values, so now we want to find all the values of x where f of x is zero.

02:25 Remember, that means where y is zero.

02:27 So if we take a look at the graph here, y is zero.

02:32 Zero along the x -axis so let's find all the values i see a point here so x is zero four and six they're all your x intercepts so when x is zero four and six now let's take a look at part f here for what values of x is f of x less than zero which means y is less than zero which means we want y to be negative numbers or sorry the x values should be that means y is negative so let's look at the graph so i'm going to draw a line all the values below the x -axis okay is where y is negative so it wanted the x values so as you could see here from zero to four not including zero to four those values will be negative.

03:46 So we can write that in interval notation...

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