State whether each of the following functions specifies a direct or an inverse relationship.

a. y = 5x

b. y = 10 - 2x

c. y = 3 + x

d. y = −3x

A direct relationship means there is a positive relationship between two variables meaning an increase in one will increase the other and vice versa.

An inverse relationship means there is a negative relationship between two variables meaning an increase in one will decrease the other and vice versa.

The function shows a direct relationship between the two variables as the value of one variable $x$increases the value of $y$also increases. It is shown using an example,

$$\begin{gathered} x=1 \\ y=5\times 1 \\ \Rightarrow y=5 \\ x=3 \\ y=5\times 3 \\ \Rightarrow y=15 \end{gathered}$$

The function shows an inverse relationship between the two variables as the value of one variable $x$increases the value of $y$also decreases. It is shown using an example,

$$\begin{gathered} x=1 \\ y=10-2\times 1 \\ \Rightarrow y=10-2 \\ \Rightarrow y=8 \\ x=3 \\ y=10-2\times 3 \\ \Rightarrow y=10-6 \\ \Rightarrow y=4 \end{gathered}$$

The function shows a direct relationship between the two variables as the value of one variable $x$increases the value of $y$also increases. It is shown using an example,

$$\begin{gathered} x=1 \\ y=3+1 \\ \Rightarrow y=4 \\ x=3 \\ y=3+3 \\ \Rightarrow y=6 \end{gathered}$$

The function shows an inverse relationship between the two variables as the value of one variable $x$increases the value of $y$also decreases. It is shown using an example,

$$\begin{gathered} x=1 \\ y=-3\times 1 \\ \Rightarrow y=-3 \\ x= \\ y=-3\times 3 \\ \Rightarrow y=-9 \end{gathered}$$