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1.3 Graphs of Functions

Contrasting Examples

In your first course in algebra, you learned to graph lines, and possibly other types of graphs, such as parabolas and circles. Consider the following two equations:

Equation 1: \(y=x^2\)

Equation 2: \(y^2=x\)

Exercise 1.3.1

Explain why, in the first equation, \(y\) is a function of \(x\), and why, in the second equation \(y\) is not a function of \(x\).

See Solution

Exercise 1.3.2

Sketch the graphs of \(y=x^2\) and \(y^2=x\). Graph paper

See Solution

The Vertical Line Test

One may use the graph of a function to estimate the output for any given input.

Look at your graph of equation 1 and select some number on the \(x\)-axis as the input into the function \(y=x^2\). Label the input number \(x\).

Now draw a vertical line through the number you selected.

At the point where the vertical line intersects the graph, draw a horizontal line.

Look at where the horizontal line crosses the \(y\)-axis. The number where the horizontal line crosses the \(y\)-axis is the output corresponding to the input you selected on the \(x\)-axis. Label the output number \(y\).

The graph of every function operates in this way.

Now try to apply the same process to the graph of the second equation \(y^2=x\), which, remember, is not a function.

Notice that most vertical lines intersect the graph in more than one place. Thus, one must draw two horizontal lines that intersect the graph in two places. So this cannot be the graph of a function, since a function has only one output for any given input.

These two examples illustrate an important principle:

No vertical line intersects the graph of a function more than once.

This principle is sometimes referred to as the vertical line test.