Question

1)Why is it impossible for a function to have fewer values in its domain than in its range?

Answer 1

Part (1):

For a relation to be called a function, each input must have one and only one corresponding output.

Now, we know that the input of the relation is known as its domain while the output is known as its range.

Based on the above definition, each input (value from domain) has only one output (value from range)

Therefore, the values for domain and range are equal, otherwise, the relation won't be a function

Part (2):

The graph of the function is shown in the attached image.

Now, a linear function:

1- is a straight line. This means that it has neither curved nor sharped bends.

In the graph of the given function, we can see that it has a sharp bend at the point (5,0)

2- has a constant slope all over the graph.

Let's test this condition on our graph:

For the two points (0,5) and (1,4):

slope = $\frac{4-5}{1-0} = -1$

For the two points (6,1) and (7,2):

slope = $\frac{2-1}{7-6} = 1$

We can see that the slope is not constant along the whole graph.

Based on the above two conditions, the absolute function is not a linear function

Part(3):

There are two types of proportionality:

1- Direct proportionality: This means that as one value increases, the other would increase at the same rate

Let's check this on the given function:

f (x) = -3x

At x = 1 ..........> f (x) = -3(1) = -3

At x = 2 .........> f (x) = -3(2) = -6

We can note that the condition of direct proportionality is not fulfilled, therefore, the relation is not showing direct proportionality

2- Inverse proportionality: This means that as one value increases, the other would decrease by the same rate.

Inverse proportionality has the general formula: y = $\frac{k}{x}$

where k is the constant of proportionality.

The given function is: f (x) = -3x

We can note that it does not have the same format as the general formula of the inverse proportionality.

Therefore, the given relation does not show inverse proportionality

Based on the above, we can conclude that the given relation is non-proportional

Part (4):

The graphs are shown in the second attached image

Let's check the givens:

1) y = 9

We know that this would be a line parallel to the x-axis constant at y = 9

Therefore, this choice is correct

2) -2x² + 4

We know that the second degree polynomial gives a curve not a line.

Therefore, this choice is incorrect

3) y = -8 + 3

The general form of the linear line is y = mx + b

The given equation has the same format as the general one.

Therefore, this choice is correct

4) y = 7/x

Again, the general form of the linear line is y = mx + b

The given equation has the a different format than the general one.

Therefore, this choice is incorrect

Hope this helps :)