How to find the domain and the range of a function given its graph (example) | Khan Academy

Summary

The video transcript explains how to determine the domain and range of a function f(x) by analyzing its graph. The speaker walks through several examples, showing how to identify where the function is defined (domain) and what y-values it can take (range). For domain, they explain that f(x) is defined only within certain x-values, while for range, they discuss the possible y-values that f(x) can reach. The process involves understanding inequalities for both the x and y values, providing clarity through practical graph examples.

Takeaways

• 📉 The function f(x) is graphed, and we need to determine its domain.
• 🔢 The function starts being defined at x = -6, and f(-6) equals 5.
• 📏 The domain of the function is the set of x values from -6 to 7, inclusive.
• 📈 The function f(x) is defined for any x value that satisfies -6 ≤ x ≤ 7.
• 🔍 Another example shows that f(x) is not defined until x = -1, where f(-1) equals -5.
• 🧮 For the second example, the domain is -1 ≤ x ≤ 7.
• 🔢 The range of the function is determined by the set of possible y values.
• 📊 The lowest y value for f(x) is 0, and the highest is 8, giving a range of 0 ≤ f(x) ≤ 8.
• ✏️ A third example shows a domain of -2 ≤ x ≤ 5.
• 🔍 In each case, the function’s domain and range are determined based on where the function is defined and the set of corresponding y values.

Q & A

What is the domain of the function f(x) in the first example?

The domain of the function is from x = -6 to x = 7, inclusive. This means the function is defined for all values of x where -6 ≤ x ≤ 7.

What does it mean when the function is defined between certain x-values?

It means that for any x-value within the specified range, you can find a corresponding y-value (or f(x)). Outside this range, the function is not defined, meaning there is no corresponding y-value.

What is the value of f(x) when x = -6 in the first example?

When x = -6, the value of f(x) is 5.

How do you determine if a function is defined for a specific x-value?

You check the graph to see if there is a point corresponding to that x-value. If there is, the function is defined for that value; if not, the function is undefined for that x-value.

In the second example, what is the domain of f(x)?

In the second example, the domain of f(x) is from x = -1 to x = 7, inclusive. This means the function is defined for all x-values where -1 ≤ x ≤ 7.

What is the range of the function f(x) in the third example?

The range of the function is from 0 to 8, inclusive. This means the function takes on values of y (or f(x)) between 0 and 8, including these values.

How do you find the range of a function from its graph?

To find the range, you look at the y-values that the function takes on across the entire domain. The lowest y-value is the minimum of the range, and the highest y-value is the maximum.

What is the significance of the y-values in the range of a function?

The y-values in the range represent all possible outputs (f(x) values) that the function can produce based on the x-values in the domain.

In the final example, what is the domain of f(x)?

The domain of the function in the final example is from x = -2 to x = 5, inclusive. This means the function is defined for all x-values between -2 and 5.

How does the domain differ from the range of a function?

The domain represents all possible x-values for which the function is defined, while the range represents all possible y-values (f(x)) that the function can output.

Outlines

00:00

📉 Understanding the Domain of the Function

This paragraph explains how to identify the domain of the function f(x) from its graph. The graph shows that the function is undefined for values less than -6, and starts being defined at x = -6 where f(x) = 5. The function continues to be defined for values of x up to and including x = 7, where f(x) remains 5. Therefore, the domain of the function is x values between -6 and 7, inclusive. The explanation concludes by stating that for any x within this range, the function is defined.

🧐 Exploring Another Function’s Domain

This section presents another example of determining the domain of a graphed function. The function is not defined for x values smaller than -1, but it starts being defined at x = -1 where f(x) = -5. It remains defined up to x = 7, where the function includes this value. The domain is thus from x = -1 to x = 7, inclusive, and the function is defined for any x that satisfies this inequality. The explanation uses a similar reasoning as in the previous example.

📊 Investigating the Range of the Function

Here, the focus shifts from the domain to the range of the function, which refers to the set of possible y-values. The graph shows that the lowest value the function reaches is y = 0, which occurs at f(-4). The highest value is y = 8, seen at f(7). Therefore, the range of the function is all y-values from 0 to 8, inclusive. This explanation highlights how the range differs from the domain by focusing on the vertical axis (y-values) instead of the horizontal axis (x-values).

🔍 Determining the Domain of Another Function

The final paragraph goes over the domain of a new function, which is defined for x values between -2 and 5, inclusive. For each x within this interval, the function has a corresponding y-value shown on the graph, such as f(-2) = -4 and f(-1) = -3. This segment emphasizes that the domain includes not only integer values but also any x-value between -2 and 5, following the same reasoning as previous examples.

Keywords

💡Domain

In mathematics, the domain of a function is the set of all possible input values (x-values) for which the function is defined. In the video, the speaker explains that the function f(x) is only defined for x values between -6 and 7, inclusive, meaning the domain of this function is from -6 to 7. The domain concept is central to understanding where a function exists on a graph.

💡Range

The range of a function is the set of all possible output values (y-values) that the function can produce. In the video, the speaker points out that the range of the function is from 0 to 8, meaning the function’s output lies between these values. This concept helps identify the vertical span of the function on a graph.

💡Function

A function is a relationship that maps each input (x) to exactly one output (y). The video revolves around examining f(x), a function whose inputs and outputs are plotted on a graph. By exploring the function’s domain and range, the speaker demonstrates how functions behave on a graph and how to interpret their values.

💡Graph

A graph in this context visually represents the function f(x), showing the relationship between input (x) and output (y). The video extensively discusses the graph of f(x), which helps viewers understand where the function is defined and its corresponding outputs. The graph is a crucial visual tool for exploring the function’s domain and range.

💡Inequality

Inequalities are mathematical expressions that show the relationship between two values, stating whether one is greater than, less than, or equal to another. In the video, inequalities are used to define the domain (e.g., -6 ≤ x ≤ 7) and range (e.g., 0 ≤ f(x) ≤ 8), specifying the intervals where the function is valid.

💡x-value

An x-value represents the input of the function, which is plotted on the horizontal axis of a graph. In the video, the x-values range from -6 to 7, and the speaker explains how to check if a particular x-value has a corresponding output (f(x)). Understanding x-values is crucial for determining the domain of the function.

💡y-value

A y-value, or f(x), represents the output of the function for a given x-value, plotted on the vertical axis of the graph. The video discusses how to find the y-value by checking the height of the graph at specific x-values, which contributes to understanding the range of the function.

💡f(x)

f(x) is the notation used to represent a function in terms of x. It signifies that the function’s output depends on the input x. The video analyzes the behavior of f(x) at different points, explaining how it is defined and how to read its values from the graph. The exploration of f(x) is fundamental to grasping the video's overall message.

💡Defined

In mathematics, a function is said to be defined for certain x-values if there is a corresponding y-value (output) for those inputs. In the video, the function f(x) is defined for x-values between -6 and 7. This term is used to explain where the function exists on the graph and where it can be evaluated.

💡Double inequality

A double inequality involves two inequality signs, used to describe the range or domain of a function in a concise manner. For example, in the video, the domain is expressed as -6 ≤ x ≤ 7, meaning x is between -6 and 7. This helps communicate the set of x-values for which the function is defined in an efficient way.

Highlights

The function f(x) is defined for x values between -6 and 7, inclusive.

At x = -6, f(x) equals 5, indicating the starting point of the graph.

At x = 7, f(x) also equals 5, marking the endpoint of the graph.

The domain of the function is the interval [-6, 7].

To find the function value for a specific x, you move up from the x-axis to the point on the graph.

For a second function, the domain is from x = -1 to x = 7.

At x = -1, f(x) equals -5, marking the start of the second function's domain.

The domain of the second function is the interval [-1, 7].

The range of the first function is from y = 0 to y = 8.

The lowest value of f(x) is 0, which occurs at x = -4.

The highest value of f(x) is 8, which occurs at x = 7.

The function's range is the interval [0, 8].

The domain of a third function is from x = -2 to x = 5.

At x = -2, f(x) equals -4, marking the start of the third function's domain.

The function is defined for all x in the interval [-2, 5], with corresponding y values obtainable from the graph.

Transcripts

The function f of x is graphed.

What is its domain?

So the way it's graphed right over here,

we could assume that this is the entire function

definition for f of x.

So for example, if we say, well, what

does f of x equal when x is equal to negative 9?

Well, we go up here.

We don't see it's graphed here.

It's not defined for x equals negative 9 or x equals

negative 8 and 1/2 or x equals negative 8.

It's not defined for any of these values.

It only starts getting defined at x equals negative 6.

At x equals negative 6, f of x is equal to 5.

And then it keeps getting defined. f of x

is defined for x all the way from x equals

negative 6 all the way to x equals 7.

When x equals 7, f of x is equal to 5.

You can take any x value between negative 6,

including negative 6, and positive 7,

including positive 7, and you just

have to see-- you just have to move up

above that number, wherever you are,

to find out what the value of the function is at that point.

So the domain of this function definition?

Well, f of x is defined for any x that

is greater than or equal to negative 6.

Or we could say negative 6 is less than or equal to x,

which is less than or equal to 7.

If x satisfies this condition right over here,

the function is defined.

So that's its domain.

So let's check our answer.

Let's do a few more of these.

The function f of x is graphed.

What is its domain?

Well, exact similar argument.

This function is not defined for x is negative 9, negative 8,

all the way down or all the way up I should say to negative 1.

At negative 1, it starts getting defined.

f of negative 1 is negative 5.

So it's defined for negative 1 is less than or equal to x.

And it's defined all the way up to x equals 7,

including x equals 7.

So this right over here, negative 1

is less than or equal to x is less than or equal to 7,

the function is defined for any x that

satisfies this double inequality right over here.

Let's do a few more.

The function f of x is graphed.

What is its range?

So now, we're not thinking about the x's

for which this function is defined.

We're thinking about the set of y values.

Where do all of the y values fall into?

Well, let's see.

The lowest possible y value or the lowest possible value

of f of x that we get here looks like it's 0.

The function never goes below 0.

So f of x-- so 0 is less than or equal to f of x.

It does equal 0 right over here. f of negative 4 is 0.

And then the highest y value or the highest value

that f of x obtains in this function definition is 8.

f of 7 is 8.

It never gets above 8, but it does equal 8 right over here

when x is equal to 7.

So 0 is less than f of x, which is less than or equal to 8.

So that's its range.

Let's do a few more.

This is kind of fun.

The function f of x is graphed.

What is its domain?

So once again, this function is defined for negative 2.

Negative 2 is less than or equal to x, which is less than

or equal to 5.

If you give me an x anywhere in between negative 2 and 5,

I can look at this graph to see where the function is defined.

f of negative 2 is negative 4.

f of negative 1 is negative 3.

So on and so forth, and I can even

pick the values in between these integers.

So negative 2 is less than or equal to x, which is less than

or equal to 5.