A relation is a function, if and only if, it passes the Vertical Line Test (VLT)—that is, no vertical line exists that passes through its graph more than once. From the diagram below, we see that no such line exists:

The relation passes the VLT, so it is a function.

A function has an inverse if and only if it passes the Horizontal Line Test (HLT) - that is, no horizontal line exists that passes through its graph more than once. From the diagram below, we see that no such line exists:

The function passes the HLT, so it has an inverse.

A relation is a function, if and only if, it passes the Vertical Line Test (VLT)—that is, no vertical line exists that passes through its graph more than once. From the diagram below, we see that no such line exists;

The relation passes the VLT, so it is a function.

A function has an inverse if and only if it passes the Horizontal Line Test (HLT) - that is, no horizontal line exists that passes through its graph more than once. From the diagram below, we see at least one such line exists.

The function fails the HLT, so it does not have an inverse.

To find the inverse of a linear function, switch the variables and solve for y.

$\displaystyle y=\frac{x-3}{2}$

Switch the variables:

$\displaystyle x=\frac{y-3}{2}$

Multiply both sides by 2:

$\displaystyle 2x=y-3$

Add 3 to both sides to isolate y:

Interchange the x and y-variables.

$\displaystyle x=2(5-y)$

Solve for y.  Divide by two on both sides.

$\displaystyle \frac{x}{2}=\frac{2(5-y)}{2}$

$\displaystyle \frac{x}{2}=5-y$

Add $\displaystyle y$ on both sides.

$\displaystyle \frac{x}{2}+y=5-y+y$

$\displaystyle \frac{x}{2}+y=5$

Subtract $\displaystyle \frac{x}{2}$ on both sides.

$\displaystyle \frac{x}{2}+y-\frac{x}{2}=5-\frac{x}{2}$

Simplify both sides.

The answer is:  

A horizontal line has infinitely many values for $\displaystyle x$, but only one possible value for $\displaystyle y$. Thus, it is always of the form $\displaystyle y = c$, where $\displaystyle c$ is a constant. Horizontal lines have a slope of $\displaystyle 0$. The only equation of this form is $\displaystyle y = 0$. 

The equation $\displaystyle x=5$ is a vertical line, so the perpendicular line must be horizontal. The only answer choice that is a horizontal line is $\displaystyle y = -5$.

A vertical line has infinitely many values of $\displaystyle y$ but only one value of $\displaystyle x$. Thus, vertical lines are of the form $\displaystyle x = c$, where $\displaystyle c$ is a real number. The only equation of this form is $\displaystyle x = 2$. 

The graph of x=5 is a vertical line. The equation x=5 represents all points with x- value equal to 5.

Try to plot a couple of points with an x-value of 5.

A few examples are (5, 0), (5, 2), (5,5).

Draw a line connecting the points and you obtain a vertical line intercepting the x-axis at (5,0).

Think about the meaning of a vertical line on the coordinate grid. The $\displaystyle y$ value changes to any value, yet the $\displaystyle x$ value always stays the same. Thus, we are talking about an equation in which the $\displaystyle y$ is free, or is not effected, and the $\displaystyle x$ is constant. This is an equation of the form $\displaystyle x = c$, where $\displaystyle c$ is a constant. 

Think about what it means to be a horizontal line. The $\displaystyle x$ value changes to be any real number, but the $\displaystyle y$ value always remains constant. Thus, we are looking for an equation in which the $\displaystyle y$ value is constant and the  $\displaystyle x$ value is not present. This would be any equation of the form $\displaystyle y = c$, where $\displaystyle c$ is a constant.