Functions are packages containing three sets: domain, codomain, and pairs. The pairs connect the domain values to codomain values and it is this relationship we want to investigate.

For us, the codomain is often \(\mathbb {R}\) and so we often ignore this until it becomes important. (There are many reasons for it to become important). Thus, we often think of our real-valued functions as three sets: domain, range, and pairs.

We would like to know how the range values are affected by the domain values.

• We are interested in the function values - the range values. But they are controlled by the domain values.
  
• We ask questions about the function values, but the answers are in the domain.

A graph provides a global view of all of the pairs, which reveals many of the patterns, features, and characteristics we seek.

However, we must separate the graph from the function. The graph helps us answer the questions, but the graph doesn’t hold the answers. The answers are in the domain and range. The answers are not the points in the graph. The answers are the numbers in the domain and range.

The graph displays patterns in its points. These patterns help us analyze the function which has values in the range connected to numbers in the domain.

Maximum

Below is the graph of \(y=f(x)\).

The highest point on the graph of \(f\) is

\((-5,0)\) \((-2,9)\) \((0,5)\) \((-1,-2)\) \((5,-6)\) \((8,0)\)

The maximum value of \(f\) is \(\answer {9}\), which occurs at \(\answer {-2}\).

Below is the graph of \(y=g(t)\).

There is no highest point on the graph. Therefore, \(g\) has no maximum value.

Minimum

Below is the graph of \(y=f(x)\).

The lowest point on the graph of \(f\) is

\((-5,0)\) \((-2,9)\) \((0,5)\) \((-1,-2)\) \((5,-6)\) \((8,0)\)

The minimum value of \(f\) is \(\answer {-6}\), which occurs at \(\answer {5}\).

Below is the graph of \(y=H(k)\).

There is no lowest point on the graph. Therefore, \(H\) has no minimum value.