Given the following linear function sketch the graph of the function and find the domain and range.

ƒ(x) = -3x + 7

Unless the line is vertical or horizontal, the domain and range are both all real numbers.

To sketch the line, pick two values of x, say 0 and 1, find the corresponding values of f(x), and then plot the points and draw the line through them.

Sure, let's start with a joke before we get into the math!

Why don't scientists trust atoms?

Because they make up everything!

Now to the math. For the linear function ƒ(x) = -3x + 7:

To sketch the graph, we need to plot a few points. One easy point to start with is the y-intercept, which is the point where x = 0. So when x = 0, ƒ(x) = -3(0) + 7 = 7. This gives us the point (0, 7).

Next, we can find another point by choosing any value for x and plugging it into the equation. Let's choose x = 2. When x = 2, ƒ(x) = -3(2) + 7 = 1. So we have the point (2, 1).

Now we can plot these two points on a coordinate plane and draw a straight line through them.

As for the domain and range:

The domain is all the possible x-values for the function. Since this is a linear function, the domain is all real numbers (-∞, ∞).

The range is all the possible y-values for the function. Looking at the graph, we can see that the y-values go from negative infinity to positive infinity, so the range is also all real numbers (-∞, ∞).

I hope that helps! Let me know if you have any other questions or if you want another joke!

To sketch the graph of the linear function ƒ(x) = -3x + 7, you can follow these steps:

Step 1: Label the axes of the coordinate plane. Let the horizontal axis represent the x-axis, and let the vertical axis represent the y-axis.

Step 2: Find two points that lie on the line. To find the first point, substitute x = 0 into the function and solve for y:
ƒ(0) = -3(0) + 7 = 7
So, the first point is (0, 7).

To find the second point, substitute x = 1 into the function and solve for y:
ƒ(1) = -3(1) + 7 = 4
So, the second point is (1, 4).

Step 3: Plot the two points on the coordinate plane.

Step 4: Draw a straight line passing through the two points. This line represents the graph of the linear function ƒ(x) = -3x + 7.

Now, let's find the domain and range of the function:

The domain of a linear function is all the possible values for x. Since there are no restrictions on the possible values of x in the given function, the domain is all real numbers, or (-∞, +∞).

The range of a linear function is all the possible values for y. From the graph, it can be observed that as x increases, y decreases. Therefore, the range is also all real numbers, or (-∞, +∞).

In summary:
- The graph of the function ƒ(x) = -3x + 7 is a straight line passing through the points (0, 7) and (1, 4).
- The domain of the function is all real numbers, or (-∞, +∞).
- The range of the function is also all real numbers, or (-∞, +∞).

To sketch the graph of the linear function ƒ(x) = -3x + 7 and find the domain and range, we can use a few steps.

Step 1: Graph the function:
To sketch the graph, we need to plot a few points and connect them with a straight line. We can choose any values for x and find the corresponding y-values.

Let's choose three values for x: -1, 0, and 1.

For x = -1:
ƒ(-1) = -3(-1) + 7 = 10
So, we have the point (-1, 10).

For x = 0:
ƒ(0) = -3(0) + 7 = 7
So, we have the point (0, 7).

For x = 1:
ƒ(1) = -3(1) + 7 = 4
So, we have the point (1, 4).

Now, plot these three points (-1, 10), (0, 7), and (1, 4) on a coordinate plane. Once you have plotted these points, draw a straight line passing through them. This line represents the graph of the function ƒ(x) = -3x + 7.

Step 2: Find the domain:
The domain is the set of all possible input values (values of x) for which the function is defined. In this case, there are no restrictions on x, so the domain of the function ƒ(x) = -3x + 7 is all real numbers, or (-∞, +∞).

Step 3: Find the range:
The range is the set of all possible output values (values of y) for which the function produces a corresponding y-value. Since the graph is a straight line that continues indefinitely in both positive and negative directions, the range of the function ƒ(x) = -3x + 7 will also be all real numbers, or (-∞, +∞).