It is important to find ways to examine general properties of functions efficiently.

In This Module

We will look at five base graphs.

For each of these functions, we will look at efficient ways to sketch their graph, discuss domain and range, and make observations about some characteristics or features of each graph.

For the exponential function, we will look at a specific case: \(f(x)=2^x\).

Goal: Sketch the graph of \(f(x)=x^2\). State the domain and range. List any other features of the graph.

A common method for sketching any function is to use a table of values.

We could use any values for \(x\), but choosing convenient
values is a good idea.

Plot the points and then join them with a smooth curve.

Since it is possible to square any real number, the domain is \(\{x \mid x \in \mathbb{R}\}\).

When any real number is squared, the result will be greater than or equal to \(0\).

Therefore, the range is \(\{y \mid y\ge 0, y \in \mathbb{R}\}\).

The graph is symmetric about the \(y\)-axis.

It has a minimum point which occurs at the vertex \((0,0)\).

Goal: Sketch the graph of \(f(x)=\sqrt{x}\). State the domain and range. List any other features of the graph.

Here is a table of values.

The square root of a negative number does not give a real number.

Thus, we use values for \(x\ge 0\), but choosing convenient values is a good idea.

Plot the points and then join them with a smooth curve.

Since the square root of a negative number does not give a real number, the domain is \(\{x \mid x\ge0,x \in \mathbb{R}\}\).

The range is \(\{y \mid y\ge 0, y \in \mathbb{R}\}\).

The graph has a definite starting point at \((0,0)\).

The graph also appears to be related to the function \(y=x^2\).

This idea will be pursued in another module.