Use the given graph of the function f to answer parts (a)-(n). (a) Find f(0) and f(6). (b) Find

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

Use the given graph of the function $f$ to answer parts $(\mathrm{a})-(\mathrm{n})$. (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? (I) 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 $(\mathrm{a})-(\mathrm{n})$.
(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?
(I) 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

Function Evaluation

Function evaluation involves determining the output of the function for a given input by locating the corresponding point on its graph. This concept is fundamental in determining specific function values and is often carried out by reading the y-coordinate at a given x-coordinate.

Graph Interpretation

Graph interpretation is the process of extracting information about a function by analyzing its plotted curve. It includes identifying the shape, key points, where the function is above or below the x-axis, and any intersections with axes or other lines.

Domain

The domain refers to the set of all possible input values (x-values) for which the function is defined. When observed graphically, it is represented by the span of the x-values over which the function's graph extends.

Range

The range is the set of all output values (y-values) that a function can produce. This is directly determined from the graph by identifying the lowest and highest points attained by the function.

Intercepts

Intercepts are the points where the graph crosses the axes. The x-intercepts occur when the output value is zero (f(x) = 0), while the y-intercept occurs when the input value is zero. These points are critical to understanding the function's behavior.

Sign Analysis

Sign analysis involves determining whether the function's value is positive or negative at certain intervals. By observing the graph, one can tell if the function lies above the x-axis (positive values) or below the x-axis (negative values), which is useful in solving inequalities.

Intersection with Horizontal and Vertical Lines

This concept refers to identifying where the graph meets specific lines, such as horizontal lines defined by y = constant or vertical lines defined by x = constant. These intersections are essential for solving equations like f(x) = constant and for understanding the function's behavior relative to given reference lines.

Transcript

00:01 So in this problem, we're going to answer some questions about this graph.

00:04 So we're analyzing the graph, and we're going to determine a bunch of different values and portions of the graph that this problem asks about.

00:12 So let's begin with a.

00:17 So a asks us to find f of 0 and f of 6.

00:23 So this means find what the y value equals when x is equal to 0.

00:30 So when x is equal to 0, we see that y is also equal to 0.

00:35 So we have 0 as our first answer.

00:40 F of 6 is a similar question.

00:41 What is the y value when x is equal to 6? and so again we go to the x value of 6.

00:48 We see that the y value here is 0.

00:50 So the answer here is also going to be 0.

00:54 Part b is pretty similar.

00:57 We want to find f of 2 and f of negative 2.

01:01 Let's take a look.

01:03 F of 2 will go to 2 on the x -axis, and we see that our y value is down here at negative 2.

01:10 So that is the answer here.

01:13 And f of negative 2 means we need to find where x is negative 2, which is down here.

01:19 And we see that y is 1 at that point.

01:22 So 1 is our answer here.

01:26 Next up, we need to look at f of 3.

01:30 And we don't need to find a specific value.

01:32 We just need to see if it's positive or negative.

01:34 Negative.

01:36 So we go to where x equals 3, which is right around here, and we take a look at where we end up.

01:42 Now we don't know exactly where, but we can tell this is definitely negative.

01:46 We are underneath the x -axis, so negative is our answer here.

01:55 Continuing on to part d is quite similar.

01:59 D asks us now for f of negative 1.

02:04 Again, just wanting to see if it's positive or negative.

02:08 So going to negative 1 on the x -axis, we see that our y value is something positive.

02:17 So there's our first few.

02:19 Let's keep going now.

02:22 We'll go on to e.

02:25 E asks us for what values of x is f of x equal to zero.

02:35 So here we need to find all the places.

02:39 Let's, we find all the places that our y value is zero and find what x values get us to that place.

02:47 So let's let, all right, we've got a y value of zero here, a y value of zero here, and a y value of zero here.

02:56 So the x values that get us these points are going to be zero, four, and six.

03:05 So those are our solutions.

03:07 Sometimes we'll have multiple solutions.

03:08 In this case, we have three, because there are three unique places where this function intersects the x -axis.

03:19 Part f is a little similar, and it asks for x when f of x is less than zero.

03:28 So in what places are we underneath the x -axis or have negative y values? and we have it here.

03:37 This is the place where our x values or our y values are negative.

03:44 So our y values are negative here, but we can't list every single x value, because there are infinitely many.

03:51 So we have to write an interval.

03:53 We say x is between 0 and 4, or 0 is less than x, which is less than 4.

04:02 That means that if x is a number that is greater than 0, but less than 4, the y value is going to be negative.

04:09 And we can see that's true.

04:13 Our next questions g and and h are about domain and range.

04:29 Domain and range ask us about all the possible x values or all the possible y values of a function.

04:35 The domain asks us about the x values.

04:38 So we need to list what x values are possible in this function.

04:43 We have quite a lot of x values, but they don't go past a certain point.

04:48 We don't go below negative 4 in terms of x...

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