The vertical line test can be used to determine if an equation is a function. In order to be a function, there must only be one  (or ) value for each value of . The vertical line test determines how many  (or ) values are present for each value of . If a single vertical line passes through the graph of an equation more than once, it is not a function. If it passes through exactly once or not at all, then the equation is a function.

The horizontal line test can be used to determine if a function is one-to-one, that is, if only one  value exists for each  (or ) value. Calculating zeroes, domain, and range can be useful for graphing an equation, but they do not tell if it is a function.

Example of a function:

Example of an equation that is not a function:

THe notation  is a composite function, which means we put the inside function g(x) into the outside function f(x). Essentially, we look at the original expression for f(x) and replace each x with the value of g(x).

The original expression for f(x) is . We will take each x and substitute in the value of g(x), which is 2x-1.

We will now distribute the -2 to the 2x - 1.

We must FOIL the  term, because . 

Now we collect like terms. Combine the terms with just an x.

Combine constants.

The answer is .

means gets plugged into .

Thus .

Calculate and plug it into .

This expression is the same as saying "take the answer of and plug it into ."

First, we need to find . We do this by plugging in for in .

Now we take this answer and plug it into .

We can find the value of by replacing with .

This is our final answer.

The function  shifts a function f(x) units to the left. Conversely,  shifts a function f(x) units to the right. In this question, we are translating the graph two units to the left.

To translate along the y-axis, we use the function  or .

We are asked to find , which is the inverse of a function. 

In order to find the inverse, the first thing we want to do is replace f(x) with y. (This usually makes it easier to separate x from its function.).

Next, we will swap x and y.

Then, we will solve for y. The expression that we determine will be equal to .

Subtract 5 from both sides.

Multiply both sides by -1.

We need to raise both sides of the equation to the 1/3 power in order to remove the exponent on the right side. 

We will apply the general property of exponents which states that .

Laslty, we will subtract one from both sides.

The expression equal to y is equal to the inverse of the original function f(x). Thus, we can replace y with .

The answer is .

The inverse of requires us to interchange and and then solve for .

Then solve for :

To find the inverse of a function, exchange the and variables and then solve for .

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